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Theorem nchoicelem16 6304
Description: Lemma for nchoice 6308. Set up stratification for nchoicelem17 6305. (Contributed by SF, 19-Mar-2015.)
Assertion
Ref Expression
nchoicelem16 {t ( ≤c We NCm NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))} V
Distinct variable group:   t,m

Proof of Theorem nchoicelem16
Dummy variables n u v x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unab 3521 . . 3 ({t ¬ ≤c We NC } ∪ {t m NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))}) = {t (¬ ≤c We NC m NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))}
2 elima3 4756 . . . . . 6 (t (((2nd (1st “ {1c})) AddC ) “ ⋃1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC )) ↔ u(u 1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) u, t ((2nd (1st “ {1c})) AddC )))
3 vex 2862 . . . . . . . . . . 11 u V
43eluni1 4173 . . . . . . . . . 10 (u 1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) ↔ {u} ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ))
5 snex 4111 . . . . . . . . . . 11 {u} V
65elcompl 3225 . . . . . . . . . 10 ({u} ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) ↔ ¬ {u} (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ))
7 elimapw13 4946 . . . . . . . . . . . 12 ({u} (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) ↔ m NC {{{m}}}, {u} ( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)))
8 spacval 6282 . . . . . . . . . . . . . . . . . 18 (m NC → ( Spacm) = Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}))
98nceqd 6110 . . . . . . . . . . . . . . . . 17 (m NCNc ( Spacm) = Nc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}))
109eqeq1d 2361 . . . . . . . . . . . . . . . 16 (m NC → ( Nc ( Spacm) = uNc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) = u))
11 tccl 6160 . . . . . . . . . . . . . . . . . . . 20 (m NCTc m NC )
12 spacval 6282 . . . . . . . . . . . . . . . . . . . 20 ( Tc m NC → ( SpacTc m) = Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}))
1311, 12syl 15 . . . . . . . . . . . . . . . . . . 19 (m NC → ( SpacTc m) = Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}))
1413eleq1d 2419 . . . . . . . . . . . . . . . . . 18 (m NC → (( SpacTc m) Fin Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ))
1513nceqd 6110 . . . . . . . . . . . . . . . . . . . 20 (m NCNc ( SpacTc m) = Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}))
1615eqeq1d 2361 . . . . . . . . . . . . . . . . . . 19 (m NC → ( Nc ( SpacTc m) = ( Tc u +c 1c) ↔ Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c)))
1715eqeq1d 2361 . . . . . . . . . . . . . . . . . . 19 (m NC → ( Nc ( SpacTc m) = ( Tc u +c 2c) ↔ Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))
1816, 17orbi12d 690 . . . . . . . . . . . . . . . . . 18 (m NC → (( Nc ( SpacTc m) = ( Tc u +c 1c) Nc ( SpacTc m) = ( Tc u +c 2c)) ↔ ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))))
1914, 18anbi12d 691 . . . . . . . . . . . . . . . . 17 (m NC → ((( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc u +c 1c) Nc ( SpacTc m) = ( Tc u +c 2c))) ↔ ( Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))))
2019notbid 285 . . . . . . . . . . . . . . . 16 (m NC → (¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc u +c 1c) Nc ( SpacTc m) = ( Tc u +c 2c))) ↔ ¬ ( Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))))
2110, 20anbi12d 691 . . . . . . . . . . . . . . 15 (m NC → (( Nc ( Spacm) = u ¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc u +c 1c) Nc ( SpacTc m) = ( Tc u +c 2c)))) ↔ ( Nc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) = u ¬ ( Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))))))
22 eldif 3221 . . . . . . . . . . . . . . . 16 ({{{m}}}, {u} ( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) ↔ ({{{m}}}, {u} SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) ¬ {{{m}}}, {u} (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)))
23 snex 4111 . . . . . . . . . . . . . . . . . . . 20 {{m}} V
2423, 3opsnelsi 5774 . . . . . . . . . . . . . . . . . . 19 ({{{m}}}, {u} SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) ↔ {{m}}, u (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))
25 opelcnv 4893 . . . . . . . . . . . . . . . . . . 19 ({{m}}, u (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) ↔ u, {{m}} (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))
26 opelres 4950 . . . . . . . . . . . . . . . . . . . 20 (u, {{m}} (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) ↔ (u, {{m}} ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) u NC ))
27 opelcnv 4893 . . . . . . . . . . . . . . . . . . . . . 22 (u, {{m}} ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) ↔ {{m}}, u ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)))
28 opelco 4884 . . . . . . . . . . . . . . . . . . . . . . 23 ({{m}}, u ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) ↔ t({{m}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t t S u))
29 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {m} V
3029brsnsi1 5775 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({{m}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)tv(t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v))
3130anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (({{m}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t t S u) ↔ (v(t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u))
32 19.41v 1901 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (v((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u) ↔ (v(t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u))
3331, 32bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({{m}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t t S u) ↔ v((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u))
3433exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . 24 (t({{m}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t t S u) ↔ tv((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u))
35 excom 1741 . . . . . . . . . . . . . . . . . . . . . . . 24 (tv((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u) ↔ vt((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u))
3634, 35bitri 240 . . . . . . . . . . . . . . . . . . . . . . 23 (t({{m}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t t S u) ↔ vt((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u))
37 anass 630 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u) ↔ (t = {v} ({m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v t S u)))
3837exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . 25 (t((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u) ↔ t(t = {v} ({m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v t S u)))
39 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {v} V
40 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (t = {v} → (t S u ↔ {v} S u))
4140anbi2d 684 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (t = {v} → (({m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v t S u) ↔ ({m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v {v} S u)))
4239, 41ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . 25 (t(t = {v} ({m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v t S u)) ↔ ({m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v {v} S u))
43 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)vv ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c){m})
44 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (v ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c){m} ↔ v, {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c))
45 spacvallem1 6281 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {x, y (x NC y NC y = (2cc x))} V
4645, 29nchoicelem10 6298 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (v, {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c) ↔ v = Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}))
4743, 44, 463bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)vv = Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}))
48 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 v V
4948, 3brssetsn 4759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({v} S uv u)
5047, 49anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v {v} S u) ↔ (v = Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) v u))
5138, 42, 503bitri 262 . . . . . . . . . . . . . . . . . . . . . . . 24 (t((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u) ↔ (v = Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) v u))
5251exbii 1582 . . . . . . . . . . . . . . . . . . . . . . 23 (vt((t = {v} {m} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)v) t S u) ↔ v(v = Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) v u))
5328, 36, 523bitri 262 . . . . . . . . . . . . . . . . . . . . . 22 ({{m}}, u ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) ↔ v(v = Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) v u))
5429, 45clos1ex 5876 . . . . . . . . . . . . . . . . . . . . . . 23 Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) V
55 eleq1 2413 . . . . . . . . . . . . . . . . . . . . . . 23 (v = Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) → (v u Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u))
5654, 55ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . 22 (v(v = Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) v u) ↔ Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u)
5727, 53, 563bitri 262 . . . . . . . . . . . . . . . . . . . . 21 (u, {{m}} ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) ↔ Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u)
5857anbi1i 676 . . . . . . . . . . . . . . . . . . . 20 ((u, {{m}} ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) u NC ) ↔ ( Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u u NC ))
59 ancom 437 . . . . . . . . . . . . . . . . . . . 20 (( Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u u NC ) ↔ (u NC Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u))
6026, 58, 593bitri 262 . . . . . . . . . . . . . . . . . . 19 (u, {{m}} (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) ↔ (u NC Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u))
6124, 25, 603bitri 262 . . . . . . . . . . . . . . . . . 18 ({{{m}}}, {u} SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) ↔ (u NC Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u))
62 eqcom 2355 . . . . . . . . . . . . . . . . . . 19 ( Nc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) = uu = Nc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}))
6354eqnc2 6129 . . . . . . . . . . . . . . . . . . 19 (u = Nc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) ↔ (u NC Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u))
6462, 63bitri 240 . . . . . . . . . . . . . . . . . 18 ( Nc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) = u ↔ (u NC Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) u))
6561, 64bitr4i 243 . . . . . . . . . . . . . . . . 17 ({{{m}}}, {u} SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) ↔ Nc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) = u)
66 elimapw11c 4948 . . . . . . . . . . . . . . . . . . 19 ({{{m}}}, {u} (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c) ↔ n{{n}}, {{{m}}}, {u} ( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))))
67 oteltxp 5782 . . . . . . . . . . . . . . . . . . . . 21 ({{n}}, {{{m}}}, {u} ( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) ↔ ({{n}}, {{{m}}} SI SI TcFn {{n}}, {u} (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))))
68 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . 24 {n} V
6968, 23opsnelsi 5774 . . . . . . . . . . . . . . . . . . . . . . 23 ({{n}}, {{{m}}} SI SI TcFn ↔ {n}, {{m}} SI TcFn)
70 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . 25 n V
7170, 29opsnelsi 5774 . . . . . . . . . . . . . . . . . . . . . . . 24 ({n}, {{m}} SI TcFn ↔ n, {m} TcFn)
72 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . 24 (nTcFn{m} ↔ n, {m} TcFn)
73 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . 24 (nTcFn{m} ↔ {m}TcFnn)
7471, 72, 733bitr2i 264 . . . . . . . . . . . . . . . . . . . . . . 23 ({n}, {{m}} SI TcFn ↔ {m}TcFnn)
75 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . 24 m V
7675brtcfn 6246 . . . . . . . . . . . . . . . . . . . . . . 23 ({m}TcFnnn = Tc m)
7769, 74, 763bitri 262 . . . . . . . . . . . . . . . . . . . . . 22 ({{n}}, {{{m}}} SI SI TcFn ↔ n = Tc m)
78 opelco 4884 . . . . . . . . . . . . . . . . . . . . . . 23 ({{n}}, {u} (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC )) ↔ v({{n}}(( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC )v v((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ){u}))
79 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({{n}}(( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC )vv(( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ){{n}})
80 brres 4949 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (v(( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ){{n}} ↔ (v( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)){{n}} v NC ))
81 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (v( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)){{n}} ↔ {{n}} ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c))v)
82 brco 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ({{n}} ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c))vu({{n}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)u u S v))
8368brsnsi1 5775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ({{n}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)ut(u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t))
8483anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (({{n}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)u u S v) ↔ (t(u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v))
85 19.41v 1901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (t((u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v) ↔ (t(u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v))
8684, 85bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (({{n}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)u u S v) ↔ t((u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v))
8786exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (u({{n}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)u u S v) ↔ ut((u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v))
88 excom 1741 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (ut((u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v) ↔ tu((u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v))
89 anass 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v) ↔ (u = {t} ({n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t u S v)))
9089exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (u((u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v) ↔ u(u = {t} ({n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t u S v)))
91 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 {t} V
92 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (u = {t} → (u S v ↔ {t} S v))
9392anbi2d 684 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (u = {t} → (({n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t u S v) ↔ ({n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t {t} S v)))
9491, 93ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (u(u = {t} ({n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t u S v)) ↔ ({n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t {t} S v))
95 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ({n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)tt ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c){n})
96 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (t ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c){n} ↔ t, {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c))
9745, 68nchoicelem10 6298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (t, {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c) ↔ t = Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}))
9895, 96, 973bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ({n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)tt = Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}))
99 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 t V
10099, 48brssetsn 4759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ({t} S vt v)
10198, 100anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (({n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t {t} S v) ↔ (t = Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) t v))
10290, 94, 1013bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (u((u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v) ↔ (t = Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) t v))
103102exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (tu((u = {t} {n} ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)t) u S v) ↔ t(t = Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) t v))
10487, 88, 1033bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (u({{n}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)u u S v) ↔ t(t = Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) t v))
105 df-clel 2349 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) vt(t = Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) t v))
106104, 105bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (u({{n}} SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)u u S v) ↔ Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) v)
10781, 82, 1063bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (v( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)){{n}} ↔ Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) v)
108107anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((v( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)){{n}} v NC ) ↔ ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) v v NC ))
109 ancom 437 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) v v NC ) ↔ (v NC Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) v))
11080, 108, 1093bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (v(( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ){{n}} ↔ (v NC Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) v))
11168, 45clos1ex 5876 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) V
112111eqnc2 6129 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) ↔ (v NC Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) v))
113110, 112bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (v(( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ){{n}} ↔ v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}))
11479, 113bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({{n}}(( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC )vv = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}))
115 brres 4949 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (v((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ){u} ↔ (v(TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))){u} v Nn ))
116 brco 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (v(TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))){u} ↔ t(v(( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))t tTcFn{u}))
117 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (v(( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))tt(( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))v)
118 brun 4692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (t(( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))v ↔ (t( AddC (1st (2nd “ {1c})))v t( AddC (1st (2nd “ {2c})))v))
119 brco 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (t( AddC (1st (2nd “ {1c})))vu(t(1st (2nd “ {1c}))u u AddC v))
120 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (t(1st (2nd “ {1c}))uu(1st (2nd “ {1c}))t)
121 brres 4949 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (u(1st (2nd “ {1c}))t ↔ (u1st t u (2nd “ {1c})))
122 eliniseg 5020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (u (2nd “ {1c}) ↔ u2nd 1c)
123122anbi2i 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((u1st t u (2nd “ {1c})) ↔ (u1st t u2nd 1c))
124121, 123bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (u(1st (2nd “ {1c}))t ↔ (u1st t u2nd 1c))
125 1cex 4142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1c V
12699, 125op1st2nd 5790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((u1st t u2nd 1c) ↔ u = t, 1c)
127120, 124, 1263bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (t(1st (2nd “ {1c}))uu = t, 1c)
128127anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((t(1st (2nd “ {1c}))u u AddC v) ↔ (u = t, 1c u AddC v))
129128exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (u(t(1st (2nd “ {1c}))u u AddC v) ↔ u(u = t, 1c u AddC v))
13099, 125opex 4588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 t, 1c V
131 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (u = t, 1c → (u AddC vt, 1c AddC v))
132130, 131ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (u(u = t, 1c u AddC v) ↔ t, 1c AddC v)
133119, 129, 1323bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (t( AddC (1st (2nd “ {1c})))vt, 1c AddC v)
13499, 125braddcfn 5826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (t, 1c AddC v ↔ (t +c 1c) = v)
135 eqcom 2355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((t +c 1c) = vv = (t +c 1c))
136133, 134, 1353bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (t( AddC (1st (2nd “ {1c})))vv = (t +c 1c))
137 brco 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (t( AddC (1st (2nd “ {2c})))vu(t(1st (2nd “ {2c}))u u AddC v))
138 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (t(1st (2nd “ {2c}))uu(1st (2nd “ {2c}))t)
139 brres 4949 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (u(1st (2nd “ {2c}))t ↔ (u1st t u (2nd “ {2c})))
140 eliniseg 5020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (u (2nd “ {2c}) ↔ u2nd 2c)
141140anbi2i 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((u1st t u (2nd “ {2c})) ↔ (u1st t u2nd 2c))
142139, 141bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (u(1st (2nd “ {2c}))t ↔ (u1st t u2nd 2c))
143 2nc 6168 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2c NC
144143elexi 2868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2c V
14599, 144op1st2nd 5790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((u1st t u2nd 2c) ↔ u = t, 2c)
146138, 142, 1453bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (t(1st (2nd “ {2c}))uu = t, 2c)
147146anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((t(1st (2nd “ {2c}))u u AddC v) ↔ (u = t, 2c u AddC v))
148147exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (u(t(1st (2nd “ {2c}))u u AddC v) ↔ u(u = t, 2c u AddC v))
14999, 144opex 4588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 t, 2c V
150 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (u = t, 2c → (u AddC vt, 2c AddC v))
151149, 150ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (u(u = t, 2c u AddC v) ↔ t, 2c AddC v)
152137, 148, 1513bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (t( AddC (1st (2nd “ {2c})))vt, 2c AddC v)
15399, 144braddcfn 5826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (t, 2c AddC v ↔ (t +c 2c) = v)
154 eqcom 2355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((t +c 2c) = vv = (t +c 2c))
155152, 153, 1543bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (t( AddC (1st (2nd “ {2c})))vv = (t +c 2c))
156136, 155orbi12i 507 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((t( AddC (1st (2nd “ {1c})))v t( AddC (1st (2nd “ {2c})))v) ↔ (v = (t +c 1c) v = (t +c 2c)))
157117, 118, 1563bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (v(( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))t ↔ (v = (t +c 1c) v = (t +c 2c)))
158 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (tTcFn{u} ↔ {u}TcFnt)
1593brtcfn 6246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ({u}TcFntt = Tc u)
160158, 159bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (tTcFn{u} ↔ t = Tc u)
161157, 160anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((v(( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))t tTcFn{u}) ↔ ((v = (t +c 1c) v = (t +c 2c)) t = Tc u))
162 ancom 437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((v = (t +c 1c) v = (t +c 2c)) t = Tc u) ↔ (t = Tc u (v = (t +c 1c) v = (t +c 2c))))
163161, 162bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((v(( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))t tTcFn{u}) ↔ (t = Tc u (v = (t +c 1c) v = (t +c 2c))))
164163exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (t(v(( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))t tTcFn{u}) ↔ t(t = Tc u (v = (t +c 1c) v = (t +c 2c))))
165 tcex 6157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Tc u V
166 addceq1 4383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (t = Tc u → (t +c 1c) = ( Tc u +c 1c))
167166eqeq2d 2364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (t = Tc u → (v = (t +c 1c) ↔ v = ( Tc u +c 1c)))
168 addceq1 4383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (t = Tc u → (t +c 2c) = ( Tc u +c 2c))
169168eqeq2d 2364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (t = Tc u → (v = (t +c 2c) ↔ v = ( Tc u +c 2c)))
170167, 169orbi12d 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (t = Tc u → ((v = (t +c 1c) v = (t +c 2c)) ↔ (v = ( Tc u +c 1c) v = ( Tc u +c 2c))))
171165, 170ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (t(t = Tc u (v = (t +c 1c) v = (t +c 2c))) ↔ (v = ( Tc u +c 1c) v = ( Tc u +c 2c)))
172116, 164, 1713bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (v(TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))){u} ↔ (v = ( Tc u +c 1c) v = ( Tc u +c 2c)))
173172anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((v(TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))){u} v Nn ) ↔ ((v = ( Tc u +c 1c) v = ( Tc u +c 2c)) v Nn ))
174 ancom 437 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((v = ( Tc u +c 1c) v = ( Tc u +c 2c)) v Nn ) ↔ (v Nn (v = ( Tc u +c 1c) v = ( Tc u +c 2c))))
175115, 173, 1743bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . 25 (v((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ){u} ↔ (v Nn (v = ( Tc u +c 1c) v = ( Tc u +c 2c))))
176114, 175anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . 24 (({{n}}(( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC )v v((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ){u}) ↔ (v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) (v Nn (v = ( Tc u +c 1c) v = ( Tc u +c 2c)))))
177176exbii 1582 . . . . . . . . . . . . . . . . . . . . . . 23 (v({{n}}(( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC )v v((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ){u}) ↔ v(v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) (v Nn (v = ( Tc u +c 1c) v = ( Tc u +c 2c)))))
178 ncex 6117 . . . . . . . . . . . . . . . . . . . . . . . 24 Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) V
179 eleq1 2413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) → (v NnNc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Nn ))
180 finnc 6243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) FinNc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Nn )
181179, 180syl6bbr 254 . . . . . . . . . . . . . . . . . . . . . . . . 25 (v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) → (v Nn Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin ))
182 eqeq1 2359 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) → (v = ( Tc u +c 1c) ↔ Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c)))
183 eqeq1 2359 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) → (v = ( Tc u +c 2c) ↔ Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))
184182, 183orbi12d 690 . . . . . . . . . . . . . . . . . . . . . . . . 25 (v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) → ((v = ( Tc u +c 1c) v = ( Tc u +c 2c)) ↔ ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))))
185181, 184anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . 24 (v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) → ((v Nn (v = ( Tc u +c 1c) v = ( Tc u +c 2c))) ↔ ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))))
186178, 185ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . 23 (v(v = Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) (v Nn (v = ( Tc u +c 1c) v = ( Tc u +c 2c)))) ↔ ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))))
18778, 177, 1863bitri 262 . . . . . . . . . . . . . . . . . . . . . 22 ({{n}}, {u} (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC )) ↔ ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))))
18877, 187anbi12i 678 . . . . . . . . . . . . . . . . . . . . 21 (({{n}}, {{{m}}} SI SI TcFn {{n}}, {u} (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) ↔ (n = Tc m ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))))
18967, 188bitri 240 . . . . . . . . . . . . . . . . . . . 20 ({{n}}, {{{m}}}, {u} ( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) ↔ (n = Tc m ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))))
190189exbii 1582 . . . . . . . . . . . . . . . . . . 19 (n{{n}}, {{{m}}}, {u} ( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) ↔ n(n = Tc m ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))))
191 tcex 6157 . . . . . . . . . . . . . . . . . . . 20 Tc m V
192 sneq 3744 . . . . . . . . . . . . . . . . . . . . . . 23 (n = Tc m → {n} = { Tc m})
193 clos1eq1 5874 . . . . . . . . . . . . . . . . . . . . . . 23 ({n} = { Tc m} → Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}))
194192, 193syl 15 . . . . . . . . . . . . . . . . . . . . . 22 (n = Tc m Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}))
195194eleq1d 2419 . . . . . . . . . . . . . . . . . . . . 21 (n = Tc m → ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ))
196194nceqd 6110 . . . . . . . . . . . . . . . . . . . . . . 23 (n = Tc mNc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}))
197196eqeq1d 2361 . . . . . . . . . . . . . . . . . . . . . 22 (n = Tc m → ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) ↔ Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c)))
198196eqeq1d 2361 . . . . . . . . . . . . . . . . . . . . . 22 (n = Tc m → ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c) ↔ Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))
199197, 198orbi12d 690 . . . . . . . . . . . . . . . . . . . . 21 (n = Tc m → (( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)) ↔ ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))))
200195, 199anbi12d 691 . . . . . . . . . . . . . . . . . . . 20 (n = Tc m → (( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))) ↔ ( Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))))
201191, 200ceqsexv 2894 . . . . . . . . . . . . . . . . . . 19 (n(n = Tc m ( Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({n}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))) ↔ ( Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))))
20266, 190, 2013bitri 262 . . . . . . . . . . . . . . . . . 18 ({{{m}}}, {u} (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c) ↔ ( Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))))
203202notbii 287 . . . . . . . . . . . . . . . . 17 {{{m}}}, {u} (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c) ↔ ¬ ( Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c))))
20465, 203anbi12i 678 . . . . . . . . . . . . . . . 16 (({{{m}}}, {u} SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) ¬ {{{m}}}, {u} (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) ↔ ( Nc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) = u ¬ ( Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))))
20522, 204bitri 240 . . . . . . . . . . . . . . 15 ({{{m}}}, {u} ( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) ↔ ( Nc Clos1 ({m}, {x, y (x NC y NC y = (2cc x))}) = u ¬ ( Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) Fin ( Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 1c) Nc Clos1 ({ Tc m}, {x, y (x NC y NC y = (2cc x))}) = ( Tc u +c 2c)))))
20621, 205syl6rbbr 255 . . . . . . . . . . . . . 14 (m NC → ({{{m}}}, {u} ( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) ↔ ( Nc ( Spacm) = u ¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc u +c 1c) Nc ( SpacTc m) = ( Tc u +c 2c))))))
207 tceq 6158 . . . . . . . . . . . . . . . . . . . 20 ( Nc ( Spacm) = uTc Nc ( Spacm) = Tc u)
208207addceq1d 4389 . . . . . . . . . . . . . . . . . . 19 ( Nc ( Spacm) = u → ( Tc Nc ( Spacm) +c 1c) = ( Tc u +c 1c))
209208eqeq2d 2364 . . . . . . . . . . . . . . . . . 18 ( Nc ( Spacm) = u → ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) ↔ Nc ( SpacTc m) = ( Tc u +c 1c)))
210207addceq1d 4389 . . . . . . . . . . . . . . . . . . 19 ( Nc ( Spacm) = u → ( Tc Nc ( Spacm) +c 2c) = ( Tc u +c 2c))
211210eqeq2d 2364 . . . . . . . . . . . . . . . . . 18 ( Nc ( Spacm) = u → ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c) ↔ Nc ( SpacTc m) = ( Tc u +c 2c)))
212209, 211orbi12d 690 . . . . . . . . . . . . . . . . 17 ( Nc ( Spacm) = u → (( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)) ↔ ( Nc ( SpacTc m) = ( Tc u +c 1c) Nc ( SpacTc m) = ( Tc u +c 2c))))
213212anbi2d 684 . . . . . . . . . . . . . . . 16 ( Nc ( Spacm) = u → ((( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))) ↔ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc u +c 1c) Nc ( SpacTc m) = ( Tc u +c 2c)))))
214213notbid 285 . . . . . . . . . . . . . . 15 ( Nc ( Spacm) = u → (¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))) ↔ ¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc u +c 1c) Nc ( SpacTc m) = ( Tc u +c 2c)))))
215214pm5.32i 618 . . . . . . . . . . . . . 14 (( Nc ( Spacm) = u ¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))) ↔ ( Nc ( Spacm) = u ¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc u +c 1c) Nc ( SpacTc m) = ( Tc u +c 2c)))))
216206, 215syl6bbr 254 . . . . . . . . . . . . 13 (m NC → ({{{m}}}, {u} ( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) ↔ ( Nc ( Spacm) = u ¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))))
217216rexbiia 2647 . . . . . . . . . . . 12 (m NC {{{m}}}, {u} ( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) ↔ m NC ( Nc ( Spacm) = u ¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))
218 rexanali 2660 . . . . . . . . . . . 12 (m NC ( Nc ( Spacm) = u ¬ (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))) ↔ ¬ m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))
2197, 217, 2183bitrri 263 . . . . . . . . . . 11 m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))) ↔ {u} (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ))
220219con1bii 321 . . . . . . . . . 10 (¬ {u} (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) ↔ m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))
2214, 6, 2203bitri 262 . . . . . . . . 9 (u 1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) ↔ m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))
222 opelco 4884 . . . . . . . . . . 11 (u, t ((2nd (1st “ {1c})) AddC ) ↔ v(u AddC v v(2nd (1st “ {1c}))t))
223 brcnv 4892 . . . . . . . . . . . . . 14 (u AddC vv AddC u)
224 brres 4949 . . . . . . . . . . . . . . 15 (v(2nd (1st “ {1c}))t ↔ (v2nd t v (1st “ {1c})))
225 eliniseg 5020 . . . . . . . . . . . . . . . . 17 (v (1st “ {1c}) ↔ v1st 1c)
226225anbi2i 675 . . . . . . . . . . . . . . . 16 ((v2nd t v (1st “ {1c})) ↔ (v2nd t v1st 1c))
227 ancom 437 . . . . . . . . . . . . . . . 16 ((v2nd t v1st 1c) ↔ (v1st 1c v2nd t))
228226, 227bitri 240 . . . . . . . . . . . . . . 15 ((v2nd t v (1st “ {1c})) ↔ (v1st 1c v2nd t))
229125, 99op1st2nd 5790 . . . . . . . . . . . . . . 15 ((v1st 1c v2nd t) ↔ v = 1c, t)
230224, 228, 2293bitri 262 . . . . . . . . . . . . . 14 (v(2nd (1st “ {1c}))tv = 1c, t)
231223, 230anbi12i 678 . . . . . . . . . . . . 13 ((u AddC v v(2nd (1st “ {1c}))t) ↔ (v AddC u v = 1c, t))
232 ancom 437 . . . . . . . . . . . . 13 ((v AddC u v = 1c, t) ↔ (v = 1c, t v AddC u))
233231, 232bitri 240 . . . . . . . . . . . 12 ((u AddC v v(2nd (1st “ {1c}))t) ↔ (v = 1c, t v AddC u))
234233exbii 1582 . . . . . . . . . . 11 (v(u AddC v v(2nd (1st “ {1c}))t) ↔ v(v = 1c, t v AddC u))
235125, 99opex 4588 . . . . . . . . . . . 12 1c, t V
236 breq1 4642 . . . . . . . . . . . 12 (v = 1c, t → (v AddC u1c, t AddC u))
237235, 236ceqsexv 2894 . . . . . . . . . . 11 (v(v = 1c, t v AddC u) ↔ 1c, t AddC u)
238222, 234, 2373bitri 262 . . . . . . . . . 10 (u, t ((2nd (1st “ {1c})) AddC ) ↔ 1c, t AddC u)
239125, 99braddcfn 5826 . . . . . . . . . 10 (1c, t AddC u ↔ (1c +c t) = u)
240 eqcom 2355 . . . . . . . . . 10 ((1c +c t) = uu = (1c +c t))
241238, 239, 2403bitri 262 . . . . . . . . 9 (u, t ((2nd (1st “ {1c})) AddC ) ↔ u = (1c +c t))
242221, 241anbi12i 678 . . . . . . . 8 ((u 1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) u, t ((2nd (1st “ {1c})) AddC )) ↔ (m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))) u = (1c +c t)))
243 ancom 437 . . . . . . . 8 ((m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))) u = (1c +c t)) ↔ (u = (1c +c t) m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))))
244242, 243bitri 240 . . . . . . 7 ((u 1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) u, t ((2nd (1st “ {1c})) AddC )) ↔ (u = (1c +c t) m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))))
245244exbii 1582 . . . . . 6 (u(u 1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) u, t ((2nd (1st “ {1c})) AddC )) ↔ u(u = (1c +c t) m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))))
246125, 99addcex 4394 . . . . . . 7 (1c +c t) V
247 eqeq2 2362 . . . . . . . . 9 (u = (1c +c t) → ( Nc ( Spacm) = uNc ( Spacm) = (1c +c t)))
248247imbi1d 308 . . . . . . . 8 (u = (1c +c t) → (( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))) ↔ ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))))
249248ralbidv 2634 . . . . . . 7 (u = (1c +c t) → (m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))) ↔ m NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))))
250246, 249ceqsexv 2894 . . . . . 6 (u(u = (1c +c t) m NC ( Nc ( Spacm) = u → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))) ↔ m NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))
2512, 245, 2503bitri 262 . . . . 5 (t (((2nd (1st “ {1c})) AddC ) “ ⋃1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC )) ↔ m NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))
252251abbi2i 2464 . . . 4 (((2nd (1st “ {1c})) AddC ) “ ⋃1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC )) = {t m NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))}
253252uneq2i 3415 . . 3 ({t ¬ ≤c We NC } ∪ (((2nd (1st “ {1c})) AddC ) “ ⋃1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ))) = ({t ¬ ≤c We NC } ∪ {t m NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))})
254 imor 401 . . . 4 (( ≤c We NCm NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))) ↔ (¬ ≤c We NC m NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))))
255254abbii 2465 . . 3 {t ( ≤c We NCm NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))} = {t (¬ ≤c We NC m NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))}
2561, 253, 2553eqtr4i 2383 . 2 ({t ¬ ≤c We NC } ∪ (((2nd (1st “ {1c})) AddC ) “ ⋃1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ))) = {t ( ≤c We NCm NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))}
257 abexv 4324 . . 3 {t ¬ ≤c We NC } V
258 2ndex 5112 . . . . . 6 2nd V
259 1stex 4739 . . . . . . . 8 1st V
260259cnvex 5102 . . . . . . 7 1st V
261 snex 4111 . . . . . . 7 {1c} V
262260, 261imaex 4747 . . . . . 6 (1st “ {1c}) V
263258, 262resex 5117 . . . . 5 (2nd (1st “ {1c})) V
264 addcfnex 5824 . . . . . 6 AddC V
265264cnvex 5102 . . . . 5 AddC V
266263, 265coex 4750 . . . 4 ((2nd (1st “ {1c})) AddC ) V
267 ssetex 4744 . . . . . . . . . . . . 13 S V
268267ins3ex 5798 . . . . . . . . . . . . . . . . . 18 Ins3 S V
269267complex 4104 . . . . . . . . . . . . . . . . . . . . . . 23 S V
270269cnvex 5102 . . . . . . . . . . . . . . . . . . . . . 22 S V
271267cnvex 5102 . . . . . . . . . . . . . . . . . . . . . . 23 S V
27245imageex 5801 . . . . . . . . . . . . . . . . . . . . . . . . 25 Image{x, y (x NC y NC y = (2cc x))} V
273267, 272coex 4750 . . . . . . . . . . . . . . . . . . . . . . . 24 ( S Image{x, y (x NC y NC y = (2cc x))}) V
274273fixex 5789 . . . . . . . . . . . . . . . . . . . . . . 23 Fix ( S Image{x, y (x NC y NC y = (2cc x))}) V
275271, 274resex 5117 . . . . . . . . . . . . . . . . . . . . . 22 ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})) V
276270, 275txpex 5785 . . . . . . . . . . . . . . . . . . . . 21 ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))}))) V
277276rnex 5107 . . . . . . . . . . . . . . . . . . . 20 ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))}))) V
278277complex 4104 . . . . . . . . . . . . . . . . . . 19 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))}))) V
279278ins2ex 5797 . . . . . . . . . . . . . . . . . 18 Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))}))) V
280268, 279symdifex 4108 . . . . . . . . . . . . . . . . 17 ( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) V
281280, 125imaex 4747 . . . . . . . . . . . . . . . 16 (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c) V
282281complex 4104 . . . . . . . . . . . . . . 15 ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c) V
283282cnvex 5102 . . . . . . . . . . . . . 14 ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c) V
284283siex 4753 . . . . . . . . . . . . 13 SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c) V
285267, 284coex 4750 . . . . . . . . . . . 12 ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) V
286285cnvex 5102 . . . . . . . . . . 11 ( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) V
287 ncsex 6111 . . . . . . . . . . 11 NC V
288286, 287resex 5117 . . . . . . . . . 10 (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) V
289288cnvex 5102 . . . . . . . . 9 (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) V
290289siex 4753 . . . . . . . 8 SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) V
291 tcfnex 6244 . . . . . . . . . . . . 13 TcFn V
292291cnvex 5102 . . . . . . . . . . . 12 TcFn V
293292siex 4753 . . . . . . . . . . 11 SI TcFn V
294293siex 4753 . . . . . . . . . 10 SI SI TcFn V
295258cnvex 5102 . . . . . . . . . . . . . . . . . . 19 2nd V
296295, 261imaex 4747 . . . . . . . . . . . . . . . . . 18 (2nd “ {1c}) V
297259, 296resex 5117 . . . . . . . . . . . . . . . . 17 (1st (2nd “ {1c})) V
298297cnvex 5102 . . . . . . . . . . . . . . . 16 (1st (2nd “ {1c})) V
299264, 298coex 4750 . . . . . . . . . . . . . . 15 ( AddC (1st (2nd “ {1c}))) V
300 snex 4111 . . . . . . . . . . . . . . . . . . 19 {2c} V
301295, 300imaex 4747 . . . . . . . . . . . . . . . . . 18 (2nd “ {2c}) V
302259, 301resex 5117 . . . . . . . . . . . . . . . . 17 (1st (2nd “ {2c})) V
303302cnvex 5102 . . . . . . . . . . . . . . . 16 (1st (2nd “ {2c})) V
304264, 303coex 4750 . . . . . . . . . . . . . . 15 ( AddC (1st (2nd “ {2c}))) V
305299, 304unex 4106 . . . . . . . . . . . . . 14 (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c})))) V
306305cnvex 5102 . . . . . . . . . . . . 13 (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c})))) V
307292, 306coex 4750 . . . . . . . . . . . 12 (TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) V
308 nncex 4396 . . . . . . . . . . . 12 Nn V
309307, 308resex 5117 . . . . . . . . . . 11 ((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) V
310309, 289coex 4750 . . . . . . . . . 10 (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC )) V
311294, 310txpex 5785 . . . . . . . . 9 ( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) V
312125pw1ex 4303 . . . . . . . . 9 11c V
313311, 312imaex 4747 . . . . . . . 8 (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c) V
314290, 313difex 4107 . . . . . . 7 ( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) V
315287pw1ex 4303 . . . . . . . . 9 1 NC V
316315pw1ex 4303 . . . . . . . 8 11 NC V
317316pw1ex 4303 . . . . . . 7 111 NC V
318314, 317imaex 4747 . . . . . 6 (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) V
319318complex 4104 . . . . 5 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) V
320319uni1ex 4293 . . . 4 1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ) V
321266, 320imaex 4747 . . 3 (((2nd (1st “ {1c})) AddC ) “ ⋃1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC )) V
322257, 321unex 4106 . 2 ({t ¬ ≤c We NC } ∪ (((2nd (1st “ {1c})) AddC ) “ ⋃1 ∼ (( SI (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ) (( SI SI TcFn ⊗ (((TcFn (( AddC (1st (2nd “ {1c}))) ∪ ( AddC (1st (2nd “ {2c}))))) Nn ) (( S SI ∼ (( Ins3 S Ins2 ∼ ran ( S ⊗ ( S Fix ( S Image{x, y (x NC y NC y = (2cc x))})))) “ 1c)) NC ))) “ 11c)) “ 111 NC ))) V
323256, 322eqeltrri 2424 1 {t ( ≤c We NCm NC ( Nc ( Spacm) = (1c +c t) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))))} V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   wo 357   wa 358   w3a 934  wex 1541   = wceq 1642   wcel 1710  {cab 2339  wral 2614  wrex 2615  Vcvv 2859  ccompl 3205   cdif 3206  cun 3207  csymdif 3209  {csn 3737  1cuni1 4133  1cc1c 4134  1cpw1 4135   Nn cnnc 4373   +c cplc 4375   Fin cfin 4376  cop 4561  {copab 4622   class class class wbr 4639  1st c1st 4717   S csset 4719   SI csi 4720   ccom 4721  cima 4722  ccnv 4771  ran crn 4773   cres 4774  cfv 4781  2nd c2nd 4783  (class class class)co 5525  ctxp 5735   Fix cfix 5739   AddC caddcfn 5745   Ins2 cins2 5749   Ins3 cins3 5751  Imagecimage 5753   Clos1 cclos1 5872   We cwe 5895   NC cncs 6088  c clec 6089   Nc cnc 6091   Tc ctc 6093  2cc2c 6094  c cce 6096  TcFnctcfn 6097   Spac cspac 6273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-fix 5740  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-fullfun 5768  df-clos1 5873  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-tc 6103  df-2c 6104  df-ce 6106  df-tcfn 6107  df-spac 6274
This theorem is referenced by:  nchoicelem17  6305
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