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Theorem ovcelem1 6171
 Description: Lemma for ovce 6172. Set up stratification for the result. (Contributed by SF, 6-Mar-2015.)
Assertion
Ref Expression
ovcelem1 ((N V M W) → {g ab(1a N 1b M g ≈ (am b))} V)
Distinct variable groups:   a,b,g,M   N,a,b,g
Allowed substitution hints:   V(g,a,b)   W(g,a,b)

Proof of Theorem ovcelem1
Dummy variables f t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elima1c 4947 . . . 4 (g ((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) “ 1c) ↔ a{a}, g (( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c))
2 elima1c 4947 . . . . . 6 ({a}, g (( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) ↔ b{b}, {a}, g ( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )))
3 vex 2862 . . . . . . . . . . 11 g V
43otelins3 5792 . . . . . . . . . 10 ({b}, {a}, g Ins3 (( Pw1FnN) × ( Pw1FnM)) ↔ {b}, {a} (( Pw1FnN) × ( Pw1FnM)))
5 opelcnv 4893 . . . . . . . . . 10 ({b}, {a} (( Pw1FnN) × ( Pw1FnM)) ↔ {a}, {b} (( Pw1FnN) × ( Pw1FnM)))
6 opelxp 4811 . . . . . . . . . . 11 ({a}, {b} (( Pw1FnN) × ( Pw1FnM)) ↔ ({a} ( Pw1FnN) {b} ( Pw1FnM)))
7 brcnv 4892 . . . . . . . . . . . . . . 15 (t Pw1Fn {a} ↔ {a} Pw1Fn t)
8 vex 2862 . . . . . . . . . . . . . . . 16 a V
98brpw1fn 5854 . . . . . . . . . . . . . . 15 ({a} Pw1Fn tt = 1a)
107, 9bitri 240 . . . . . . . . . . . . . 14 (t Pw1Fn {a} ↔ t = 1a)
1110rexbii 2639 . . . . . . . . . . . . 13 (t N t Pw1Fn {a} ↔ t N t = 1a)
12 elima 4754 . . . . . . . . . . . . 13 ({a} ( Pw1FnN) ↔ t N t Pw1Fn {a})
13 risset 2661 . . . . . . . . . . . . 13 (1a Nt N t = 1a)
1411, 12, 133bitr4i 268 . . . . . . . . . . . 12 ({a} ( Pw1FnN) ↔ 1a N)
15 brcnv 4892 . . . . . . . . . . . . . . 15 (t Pw1Fn {b} ↔ {b} Pw1Fn t)
16 vex 2862 . . . . . . . . . . . . . . . 16 b V
1716brpw1fn 5854 . . . . . . . . . . . . . . 15 ({b} Pw1Fn tt = 1b)
1815, 17bitri 240 . . . . . . . . . . . . . 14 (t Pw1Fn {b} ↔ t = 1b)
1918rexbii 2639 . . . . . . . . . . . . 13 (t M t Pw1Fn {b} ↔ t M t = 1b)
20 elima 4754 . . . . . . . . . . . . 13 ({b} ( Pw1FnM) ↔ t M t Pw1Fn {b})
21 risset 2661 . . . . . . . . . . . . 13 (1b Mt M t = 1b)
2219, 20, 213bitr4i 268 . . . . . . . . . . . 12 ({b} ( Pw1FnM) ↔ 1b M)
2314, 22anbi12i 678 . . . . . . . . . . 11 (({a} ( Pw1FnN) {b} ( Pw1FnM)) ↔ (1a N 1b M))
246, 23bitri 240 . . . . . . . . . 10 ({a}, {b} (( Pw1FnN) × ( Pw1FnM)) ↔ (1a N 1b M))
254, 5, 243bitri 262 . . . . . . . . 9 ({b}, {a}, g Ins3 (( Pw1FnN) × ( Pw1FnM)) ↔ (1a N 1b M))
26 elrn2 4897 . . . . . . . . . 10 ({b}, {a}, g ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ tt, {b}, {a}, g ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ))
27 elin 3219 . . . . . . . . . . . 12 (t, {b}, {a}, g ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ (t, {b}, {a}, g Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) t, {b}, {a}, g Ins2 Ins2 ≈ ))
28 vex 2862 . . . . . . . . . . . . . . . . 17 t V
29 snex 4111 . . . . . . . . . . . . . . . . . 18 {b} V
30 snex 4111 . . . . . . . . . . . . . . . . . 18 {a} V
3129, 30opex 4588 . . . . . . . . . . . . . . . . 17 {b}, {a} V
3228, 31opex 4588 . . . . . . . . . . . . . . . 16 t, {b}, {a} V
3332elcompl 3225 . . . . . . . . . . . . . . 15 (t, {b}, {a} ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ↔ ¬ t, {b}, {a} (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c))
34 elima1c 4947 . . . . . . . . . . . . . . . . 17 (t, {b}, {a} (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ↔ f{f}, t, {b}, {a} ( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))))
35 elsymdif 3223 . . . . . . . . . . . . . . . . . . 19 ({f}, t, {b}, {a} ( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) ↔ ¬ ({f}, t, {b}, {a} Ins3 S {f}, t, {b}, {a} Ins2 SI3 ( Fns ⊗ ( S Image2nd ))))
3631otelins3 5792 . . . . . . . . . . . . . . . . . . . . 21 ({f}, t, {b}, {a} Ins3 S {f}, t S )
37 vex 2862 . . . . . . . . . . . . . . . . . . . . . 22 f V
3837, 28opelssetsn 4760 . . . . . . . . . . . . . . . . . . . . 21 ({f}, t S f t)
3936, 38bitri 240 . . . . . . . . . . . . . . . . . . . 20 ({f}, t, {b}, {a} Ins3 S f t)
4028otelins2 5791 . . . . . . . . . . . . . . . . . . . . 21 ({f}, t, {b}, {a} Ins2 SI3 ( Fns ⊗ ( S Image2nd )) ↔ {f}, {b}, {a} SI3 ( Fns ⊗ ( S Image2nd )))
4137, 16, 8otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . 21 ({f}, {b}, {a} SI3 ( Fns ⊗ ( S Image2nd )) ↔ f, b, a ( Fns ⊗ ( S Image2nd )))
42 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . 24 (f Fns bf, b Fns )
4337brfns 5833 . . . . . . . . . . . . . . . . . . . . . . . 24 (f Fns bf Fn b)
4442, 43bitr3i 242 . . . . . . . . . . . . . . . . . . . . . . 23 (f, b Fnsf Fn b)
45 opelco 4884 . . . . . . . . . . . . . . . . . . . . . . . 24 (f, a ( S Image2nd ) ↔ t(fImage2nd t t S a))
4637, 28brimage 5793 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (fImage2nd tt = (2ndf))
47 dfrn5 5508 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ran f = (2ndf)
4847eqeq2i 2363 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (t = ran ft = (2ndf))
4946, 48bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (fImage2nd tt = ran f)
5028, 8brsset 4758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (t S at a)
5149, 50anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((fImage2nd t t S a) ↔ (t = ran f t a))
5251exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . 24 (t(fImage2nd t t S a) ↔ t(t = ran f t a))
5337rnex 5107 . . . . . . . . . . . . . . . . . . . . . . . . 25 ran f V
54 sseq1 3292 . . . . . . . . . . . . . . . . . . . . . . . . 25 (t = ran f → (t a ↔ ran f a))
5553, 54ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . 24 (t(t = ran f t a) ↔ ran f a)
5645, 52, 553bitri 262 . . . . . . . . . . . . . . . . . . . . . . 23 (f, a ( S Image2nd ) ↔ ran f a)
5744, 56anbi12i 678 . . . . . . . . . . . . . . . . . . . . . 22 ((f, b Fns f, a ( S Image2nd )) ↔ (f Fn b ran f a))
58 oteltxp 5782 . . . . . . . . . . . . . . . . . . . . . 22 (f, b, a ( Fns ⊗ ( S Image2nd )) ↔ (f, b Fns f, a ( S Image2nd )))
59 df-f 4791 . . . . . . . . . . . . . . . . . . . . . 22 (f:b–→a ↔ (f Fn b ran f a))
6057, 58, 593bitr4i 268 . . . . . . . . . . . . . . . . . . . . 21 (f, b, a ( Fns ⊗ ( S Image2nd )) ↔ f:b–→a)
6140, 41, 603bitri 262 . . . . . . . . . . . . . . . . . . . 20 ({f}, t, {b}, {a} Ins2 SI3 ( Fns ⊗ ( S Image2nd )) ↔ f:b–→a)
6239, 61bibi12i 306 . . . . . . . . . . . . . . . . . . 19 (({f}, t, {b}, {a} Ins3 S {f}, t, {b}, {a} Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) ↔ (f tf:b–→a))
6335, 62xchbinx 301 . . . . . . . . . . . . . . . . . 18 ({f}, t, {b}, {a} ( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) ↔ ¬ (f tf:b–→a))
6463exbii 1582 . . . . . . . . . . . . . . . . 17 (f{f}, t, {b}, {a} ( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) ↔ f ¬ (f tf:b–→a))
65 exnal 1574 . . . . . . . . . . . . . . . . 17 (f ¬ (f tf:b–→a) ↔ ¬ f(f tf:b–→a))
6634, 64, 653bitrri 263 . . . . . . . . . . . . . . . 16 f(f tf:b–→a) ↔ t, {b}, {a} (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c))
6766con1bii 321 . . . . . . . . . . . . . . 15 t, {b}, {a} (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ↔ f(f tf:b–→a))
6833, 67bitri 240 . . . . . . . . . . . . . 14 (t, {b}, {a} ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ↔ f(f tf:b–→a))
693oqelins4 5794 . . . . . . . . . . . . . 14 (t, {b}, {a}, g Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ↔ t, {b}, {a} ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c))
708, 16mapval 6011 . . . . . . . . . . . . . . . 16 (am b) = {f f:b–→a}
7170eqeq2i 2363 . . . . . . . . . . . . . . 15 (t = (am b) ↔ t = {f f:b–→a})
72 abeq2 2458 . . . . . . . . . . . . . . 15 (t = {f f:b–→a} ↔ f(f tf:b–→a))
7371, 72bitri 240 . . . . . . . . . . . . . 14 (t = (am b) ↔ f(f tf:b–→a))
7468, 69, 733bitr4i 268 . . . . . . . . . . . . 13 (t, {b}, {a}, g Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ↔ t = (am b))
7529otelins2 5791 . . . . . . . . . . . . . 14 (t, {b}, {a}, g Ins2 Ins2 ≈ ↔ t, {a}, g Ins2 ≈ )
7630otelins2 5791 . . . . . . . . . . . . . 14 (t, {a}, g Ins2 ≈ ↔ t, g ≈ )
77 df-br 4640 . . . . . . . . . . . . . . 15 (tgt, g ≈ )
78 brcnv 4892 . . . . . . . . . . . . . . 15 (tggt)
7977, 78bitr3i 242 . . . . . . . . . . . . . 14 (t, g ≈ ↔ gt)
8075, 76, 793bitri 262 . . . . . . . . . . . . 13 (t, {b}, {a}, g Ins2 Ins2 ≈ ↔ gt)
8174, 80anbi12i 678 . . . . . . . . . . . 12 ((t, {b}, {a}, g Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) t, {b}, {a}, g Ins2 Ins2 ≈ ) ↔ (t = (am b) gt))
8227, 81bitri 240 . . . . . . . . . . 11 (t, {b}, {a}, g ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ (t = (am b) gt))
8382exbii 1582 . . . . . . . . . 10 (tt, {b}, {a}, g ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ t(t = (am b) gt))
84 ovex 5551 . . . . . . . . . . 11 (am b) V
85 breq2 4643 . . . . . . . . . . 11 (t = (am b) → (gtg ≈ (am b)))
8684, 85ceqsexv 2894 . . . . . . . . . 10 (t(t = (am b) gt) ↔ g ≈ (am b))
8726, 83, 863bitri 262 . . . . . . . . 9 ({b}, {a}, g ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ g ≈ (am b))
8825, 87anbi12i 678 . . . . . . . 8 (({b}, {a}, g Ins3 (( Pw1FnN) × ( Pw1FnM)) {b}, {a}, g ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ ((1a N 1b M) g ≈ (am b)))
89 elin 3219 . . . . . . . 8 ({b}, {a}, g ( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ ({b}, {a}, g Ins3 (( Pw1FnN) × ( Pw1FnM)) {b}, {a}, g ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )))
90 df-3an 936 . . . . . . . 8 ((1a N 1b M g ≈ (am b)) ↔ ((1a N 1b M) g ≈ (am b)))
9188, 89, 903bitr4i 268 . . . . . . 7 ({b}, {a}, g ( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ (1a N 1b M g ≈ (am b)))
9291exbii 1582 . . . . . 6 (b{b}, {a}, g ( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ b(1a N 1b M g ≈ (am b)))
932, 92bitri 240 . . . . 5 ({a}, g (( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) ↔ b(1a N 1b M g ≈ (am b)))
9493exbii 1582 . . . 4 (a{a}, g (( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) ↔ ab(1a N 1b M g ≈ (am b)))
951, 94bitri 240 . . 3 (g ((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) “ 1c) ↔ ab(1a N 1b M g ≈ (am b)))
9695abbi2i 2464 . 2 ((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) “ 1c) = {g ab(1a N 1b M g ≈ (am b))}
97 pw1fnex 5852 . . . . . 6 Pw1Fn V
9897cnvex 5102 . . . . 5 Pw1Fn V
99 imaexg 4746 . . . . 5 (( Pw1Fn V N V) → ( Pw1FnN) V)
10098, 99mpan 651 . . . 4 (N V → ( Pw1FnN) V)
101 imaexg 4746 . . . . 5 (( Pw1Fn V M W) → ( Pw1FnM) V)
10298, 101mpan 651 . . . 4 (M W → ( Pw1FnM) V)
103 xpexg 5114 . . . 4 ((( Pw1FnN) V ( Pw1FnM) V) → (( Pw1FnN) × ( Pw1FnM)) V)
104100, 102, 103syl2an 463 . . 3 ((N V M W) → (( Pw1FnN) × ( Pw1FnM)) V)
105 cnvexg 5101 . . . 4 ((( Pw1FnN) × ( Pw1FnM)) V → (( Pw1FnN) × ( Pw1FnM)) V)
106 ins3exg 5796 . . . 4 ((( Pw1FnN) × ( Pw1FnM)) V → Ins3 (( Pw1FnN) × ( Pw1FnM)) V)
107105, 106syl 15 . . 3 ((( Pw1FnN) × ( Pw1FnM)) V → Ins3 (( Pw1FnN) × ( Pw1FnM)) V)
108 ssetex 4744 . . . . . . . . . . . 12 S V
109108ins3ex 5798 . . . . . . . . . . 11 Ins3 S V
110 fnsex 5832 . . . . . . . . . . . . . 14 Fns V
111 2ndex 5112 . . . . . . . . . . . . . . . 16 2nd V
112111imageex 5801 . . . . . . . . . . . . . . 15 Image2nd V
113108, 112coex 4750 . . . . . . . . . . . . . 14 ( S Image2nd ) V
114110, 113txpex 5785 . . . . . . . . . . . . 13 ( Fns ⊗ ( S Image2nd )) V
115114si3ex 5806 . . . . . . . . . . . 12 SI3 ( Fns ⊗ ( S Image2nd )) V
116115ins2ex 5797 . . . . . . . . . . 11 Ins2 SI3 ( Fns ⊗ ( S Image2nd )) V
117109, 116symdifex 4108 . . . . . . . . . 10 ( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) V
118 1cex 4142 . . . . . . . . . 10 1c V
119117, 118imaex 4747 . . . . . . . . 9 (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) V
120119complex 4104 . . . . . . . 8 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) V
121120ins4ex 5799 . . . . . . 7 Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) V
122 enex 6031 . . . . . . . . . 10 V
123122cnvex 5102 . . . . . . . . 9 V
124123ins2ex 5797 . . . . . . . 8 Ins2 V
125124ins2ex 5797 . . . . . . 7 Ins2 Ins2 V
126121, 125inex 4105 . . . . . 6 ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) V
127126rnex 5107 . . . . 5 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) V
128 inexg 4100 . . . . 5 (( Ins3 (( Pw1FnN) × ( Pw1FnM)) V ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) V) → ( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) V)
129127, 128mpan2 652 . . . 4 ( Ins3 (( Pw1FnN) × ( Pw1FnM)) V → ( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) V)
130 imaexg 4746 . . . . 5 ((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) V 1c V) → (( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) V)
131118, 130mpan2 652 . . . 4 (( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) V → (( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) V)
132 imaexg 4746 . . . . 5 (((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) V 1c V) → ((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) “ 1c) V)
133118, 132mpan2 652 . . . 4 ((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) V → ((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) “ 1c) V)
134129, 131, 1333syl 18 . . 3 ( Ins3 (( Pw1FnN) × ( Pw1FnM)) V → ((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) “ 1c) V)
135104, 107, 1343syl 18 . 2 ((N V M W) → ((( Ins3 (( Pw1FnN) × ( Pw1FnM)) ∩ ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 1c) “ 1c) V)
13696, 135syl5eqelr 2438 1 ((N V M W) → {g ab(1a N 1b M g ≈ (am b))} V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∩ cin 3208   ⊕ csymdif 3209   ⊆ wss 3257  {csn 3737  1cc1c 4134  ℘1cpw1 4135  ⟨cop 4561   class class class wbr 4639   S csset 4719   ∘ ccom 4721   “ cima 4722   × cxp 4770  ◡ccnv 4771  ran crn 4773   Fn wfn 4776  –→wf 4777  2nd c2nd 4783  (class class class)co 5525   ⊗ ctxp 5735   Ins2 cins2 5749   Ins3 cins3 5751  Imagecimage 5753   Ins4 cins4 5755   SI3 csi3 5757   Fns cfns 5761   Pw1Fn cpw1fn 5765   ↑m cmap 5999   ≈ cen 6028 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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-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-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-map 6001  df-en 6029 This theorem is referenced by:  ovce  6172  fnce  6176
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