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Theorem ceex 6174
 Description: Cardinal exponentiation is stratified. (Contributed by SF, 3-Mar-2015.)
Assertion
Ref Expression
ceex c V

Proof of Theorem ceex
Dummy variables a b g m n x f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ce 6106 . . 3 c = (n NC , m NC {g ab(1a n 1b m g ≈ (am b))})
2 snex 4111 . . . . . . . . . 10 {x} V
32otelins2 5791 . . . . . . . . 9 ({{a}}, {x}, n, m Ins2 Ins3 ( S SI Pw1Fn ) ↔ {{a}}, n, m Ins3 ( S SI Pw1Fn ))
4 vex 2862 . . . . . . . . . 10 m V
54otelins3 5792 . . . . . . . . 9 ({{a}}, n, m Ins3 ( S SI Pw1Fn ) ↔ {{a}}, n ( S SI Pw1Fn ))
6 ceexlem1 6173 . . . . . . . . 9 ({{a}}, n ( S SI Pw1Fn ) ↔ 1a n)
73, 5, 63bitri 262 . . . . . . . 8 ({{a}}, {x}, n, m Ins2 Ins3 ( S SI Pw1Fn ) ↔ 1a n)
8 elimapw11c 4948 . . . . . . . . 9 ({{a}}, {x}, n, m (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c) ↔ b{{b}}, {{a}}, {x}, n, m ( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )))
9 elin 3219 . . . . . . . . . . 11 ({{b}}, {{a}}, {x}, n, m ( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ ({{b}}, {{a}}, {x}, n, m Ins2 Ins2 Ins2 ( S SI Pw1Fn ) {{b}}, {{a}}, {x}, n, m Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )))
10 snex 4111 . . . . . . . . . . . . . 14 {{a}} V
1110otelins2 5791 . . . . . . . . . . . . 13 ({{b}}, {{a}}, {x}, n, m Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ↔ {{b}}, {x}, n, m Ins2 Ins2 ( S SI Pw1Fn ))
122otelins2 5791 . . . . . . . . . . . . 13 ({{b}}, {x}, n, m Ins2 Ins2 ( S SI Pw1Fn ) ↔ {{b}}, n, m Ins2 ( S SI Pw1Fn ))
13 vex 2862 . . . . . . . . . . . . . . 15 n V
1413otelins2 5791 . . . . . . . . . . . . . 14 ({{b}}, n, m Ins2 ( S SI Pw1Fn ) ↔ {{b}}, m ( S SI Pw1Fn ))
15 ceexlem1 6173 . . . . . . . . . . . . . 14 ({{b}}, m ( S SI Pw1Fn ) ↔ 1b m)
1614, 15bitri 240 . . . . . . . . . . . . 13 ({{b}}, n, m Ins2 ( S SI Pw1Fn ) ↔ 1b m)
1711, 12, 163bitri 262 . . . . . . . . . . . 12 ({{b}}, {{a}}, {x}, n, m Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ↔ 1b m)
1813, 4opex 4588 . . . . . . . . . . . . . 14 n, m V
1918oqelins4 5794 . . . . . . . . . . . . 13 ({{b}}, {{a}}, {x}, n, m Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ {{b}}, {{a}}, {x} SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ))
20 snex 4111 . . . . . . . . . . . . . 14 {b} V
21 snex 4111 . . . . . . . . . . . . . 14 {a} V
22 vex 2862 . . . . . . . . . . . . . 14 x V
2320, 21, 22otsnelsi3 5805 . . . . . . . . . . . . 13 ({{b}}, {{a}}, {x} SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ {b}, {a}, x ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ))
24 elrn2 4897 . . . . . . . . . . . . . 14 ({b}, {a}, x ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ gg, {b}, {a}, x ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ))
25 elin 3219 . . . . . . . . . . . . . . . 16 (g, {b}, {a}, x ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ (g, {b}, {a}, x Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) g, {b}, {a}, x Ins2 Ins2 ≈ ))
2622oqelins4 5794 . . . . . . . . . . . . . . . . . 18 (g, {b}, {a}, x Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ↔ g, {b}, {a} ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c))
2720, 21opex 4588 . . . . . . . . . . . . . . . . . . 19 {b}, {a} V
28 vex 2862 . . . . . . . . . . . . . . . . . . . . . . 23 f V
2928brfns 5833 . . . . . . . . . . . . . . . . . . . . . 22 (f Fns bf Fn b)
30 brco 4883 . . . . . . . . . . . . . . . . . . . . . . 23 (f( S Image2nd )ax(fImage2nd x x S a))
3128, 22brimage 5793 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (fImage2nd xx = (2ndf))
32 dfrn5 5508 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ran f = (2ndf)
3332eqeq2i 2363 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (x = ran fx = (2ndf))
3431, 33bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . 25 (fImage2nd xx = ran f)
35 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . 26 a V
3622, 35brsset 4758 . . . . . . . . . . . . . . . . . . . . . . . . 25 (x S ax a)
3734, 36anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . 24 ((fImage2nd x x S a) ↔ (x = ran f x a))
3837exbii 1582 . . . . . . . . . . . . . . . . . . . . . . 23 (x(fImage2nd x x S a) ↔ x(x = ran f x a))
3928rnex 5107 . . . . . . . . . . . . . . . . . . . . . . . 24 ran f V
40 sseq1 3292 . . . . . . . . . . . . . . . . . . . . . . . 24 (x = ran f → (x a ↔ ran f a))
4139, 40ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . 23 (x(x = ran f x a) ↔ ran f a)
4230, 38, 413bitri 262 . . . . . . . . . . . . . . . . . . . . . 22 (f( S Image2nd )a ↔ ran f a)
4329, 42anbi12i 678 . . . . . . . . . . . . . . . . . . . . 21 ((f Fns b f( S Image2nd )a) ↔ (f Fn b ran f a))
44 df-br 4640 . . . . . . . . . . . . . . . . . . . . . 22 (f( Fns ⊗ ( S Image2nd ))b, af, b, a ( Fns ⊗ ( S Image2nd )))
45 trtxp 5781 . . . . . . . . . . . . . . . . . . . . . 22 (f( Fns ⊗ ( S Image2nd ))b, a ↔ (f Fns b f( S Image2nd )a))
4644, 45bitr3i 242 . . . . . . . . . . . . . . . . . . . . 21 (f, b, a ( Fns ⊗ ( S Image2nd )) ↔ (f Fns b f( S Image2nd )a))
47 df-f 4791 . . . . . . . . . . . . . . . . . . . . 21 (f:b–→a ↔ (f Fn b ran f a))
4843, 46, 473bitr4i 268 . . . . . . . . . . . . . . . . . . . 20 (f, b, a ( Fns ⊗ ( S Image2nd )) ↔ f:b–→a)
49 vex 2862 . . . . . . . . . . . . . . . . . . . . 21 b V
5028, 49, 35otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . 20 ({f}, {b}, {a} SI3 ( Fns ⊗ ( S Image2nd )) ↔ f, b, a ( Fns ⊗ ( S Image2nd )))
5135, 49, 28elmap 6017 . . . . . . . . . . . . . . . . . . . 20 (f (am b) ↔ f:b–→a)
5248, 50, 513bitr4i 268 . . . . . . . . . . . . . . . . . . 19 ({f}, {b}, {a} SI3 ( Fns ⊗ ( S Image2nd )) ↔ f (am b))
5327, 52releqel 5807 . . . . . . . . . . . . . . . . . 18 (g, {b}, {a} ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ↔ g = (am b))
5426, 53bitri 240 . . . . . . . . . . . . . . . . 17 (g, {b}, {a}, x Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ↔ g = (am b))
5520otelins2 5791 . . . . . . . . . . . . . . . . . 18 (g, {b}, {a}, x Ins2 Ins2 ≈ ↔ g, {a}, x Ins2 ≈ )
5621otelins2 5791 . . . . . . . . . . . . . . . . . . 19 (g, {a}, x Ins2 ≈ ↔ g, x ≈ )
57 df-br 4640 . . . . . . . . . . . . . . . . . . 19 (gxg, x ≈ )
5856, 57bitr4i 243 . . . . . . . . . . . . . . . . . 18 (g, {a}, x Ins2 ≈ ↔ gx)
59 ensym 6037 . . . . . . . . . . . . . . . . . 18 (gxxg)
6055, 58, 593bitri 262 . . . . . . . . . . . . . . . . 17 (g, {b}, {a}, x Ins2 Ins2 ≈ ↔ xg)
6154, 60anbi12i 678 . . . . . . . . . . . . . . . 16 ((g, {b}, {a}, x Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) g, {b}, {a}, x Ins2 Ins2 ≈ ) ↔ (g = (am b) xg))
6225, 61bitri 240 . . . . . . . . . . . . . . 15 (g, {b}, {a}, x ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ (g = (am b) xg))
6362exbii 1582 . . . . . . . . . . . . . 14 (gg, {b}, {a}, x ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ g(g = (am b) xg))
64 ovex 5551 . . . . . . . . . . . . . . 15 (am b) V
65 breq2 4643 . . . . . . . . . . . . . . 15 (g = (am b) → (xgx ≈ (am b)))
6664, 65ceqsexv 2894 . . . . . . . . . . . . . 14 (g(g = (am b) xg) ↔ x ≈ (am b))
6724, 63, 663bitri 262 . . . . . . . . . . . . 13 ({b}, {a}, x ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ x ≈ (am b))
6819, 23, 673bitri 262 . . . . . . . . . . . 12 ({{b}}, {{a}}, {x}, n, m Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ x ≈ (am b))
6917, 68anbi12i 678 . . . . . . . . . . 11 (({{b}}, {{a}}, {x}, n, m Ins2 Ins2 Ins2 ( S SI Pw1Fn ) {{b}}, {{a}}, {x}, n, m Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ (1b m x ≈ (am b)))
709, 69bitri 240 . . . . . . . . . 10 ({{b}}, {{a}}, {x}, n, m ( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ (1b m x ≈ (am b)))
7170exbii 1582 . . . . . . . . 9 (b{{b}}, {{a}}, {x}, n, m ( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ b(1b m x ≈ (am b)))
728, 71bitri 240 . . . . . . . 8 ({{a}}, {x}, n, m (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c) ↔ b(1b m x ≈ (am b)))
737, 72anbi12i 678 . . . . . . 7 (({{a}}, {x}, n, m Ins2 Ins3 ( S SI Pw1Fn ) {{a}}, {x}, n, m (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) ↔ (1a n b(1b m x ≈ (am b))))
74 elin 3219 . . . . . . 7 ({{a}}, {x}, n, m ( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) ↔ ({{a}}, {x}, n, m Ins2 Ins3 ( S SI Pw1Fn ) {{a}}, {x}, n, m (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)))
75 3anass 938 . . . . . . . . 9 ((1a n 1b m x ≈ (am b)) ↔ (1a n (1b m x ≈ (am b))))
7675exbii 1582 . . . . . . . 8 (b(1a n 1b m x ≈ (am b)) ↔ b(1a n (1b m x ≈ (am b))))
77 19.42v 1905 . . . . . . . 8 (b(1a n (1b m x ≈ (am b))) ↔ (1a n b(1b m x ≈ (am b))))
7876, 77bitri 240 . . . . . . 7 (b(1a n 1b m x ≈ (am b)) ↔ (1a n b(1b m x ≈ (am b))))
7973, 74, 783bitr4i 268 . . . . . 6 ({{a}}, {x}, n, m ( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) ↔ b(1a n 1b m x ≈ (am b)))
8079exbii 1582 . . . . 5 (a{{a}}, {x}, n, m ( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) ↔ ab(1a n 1b m x ≈ (am b)))
81 elimapw11c 4948 . . . . 5 ({x}, n, m (( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) “ 11c) ↔ a{{a}}, {x}, n, m ( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)))
82 breq1 4642 . . . . . . . 8 (g = x → (g ≈ (am b) ↔ x ≈ (am b)))
83823anbi3d 1258 . . . . . . 7 (g = x → ((1a n 1b m g ≈ (am b)) ↔ (1a n 1b m x ≈ (am b))))
84832exbidv 1628 . . . . . 6 (g = x → (ab(1a n 1b m g ≈ (am b)) ↔ ab(1a n 1b m x ≈ (am b))))
8522, 84elab 2985 . . . . 5 (x {g ab(1a n 1b m g ≈ (am b))} ↔ ab(1a n 1b m x ≈ (am b)))
8680, 81, 853bitr4i 268 . . . 4 ({x}, n, m (( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) “ 11c) ↔ x {g ab(1a n 1b m g ≈ (am b))})
8786releqmpt2 5809 . . 3 ((( NC × NC ) × V) (( Ins2 S Ins3 (( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) “ 11c)) “ 1c)) = (n NC , m NC {g ab(1a n 1b m g ≈ (am b))})
881, 87eqtr4i 2376 . 2 c = ((( NC × NC ) × V) (( Ins2 S Ins3 (( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) “ 11c)) “ 1c))
89 ncsex 6111 . . 3 NC V
90 ssetex 4744 . . . . . . . 8 S V
91 pw1fnex 5852 . . . . . . . . 9 Pw1Fn V
9291siex 4753 . . . . . . . 8 SI Pw1Fn V
9390, 92coex 4750 . . . . . . 7 ( S SI Pw1Fn ) V
9493ins3ex 5798 . . . . . 6 Ins3 ( S SI Pw1Fn ) V
9594ins2ex 5797 . . . . 5 Ins2 Ins3 ( S SI Pw1Fn ) V
9693ins2ex 5797 . . . . . . . . 9 Ins2 ( S SI Pw1Fn ) V
9796ins2ex 5797 . . . . . . . 8 Ins2 Ins2 ( S SI Pw1Fn ) V
9897ins2ex 5797 . . . . . . 7 Ins2 Ins2 Ins2 ( S SI Pw1Fn ) V
9990ins3ex 5798 . . . . . . . . . . . . . . 15 Ins3 S V
100 fnsex 5832 . . . . . . . . . . . . . . . . . 18 Fns V
101 2ndex 5112 . . . . . . . . . . . . . . . . . . . 20 2nd V
102101imageex 5801 . . . . . . . . . . . . . . . . . . 19 Image2nd V
10390, 102coex 4750 . . . . . . . . . . . . . . . . . 18 ( S Image2nd ) V
104100, 103txpex 5785 . . . . . . . . . . . . . . . . 17 ( Fns ⊗ ( S Image2nd )) V
105104si3ex 5806 . . . . . . . . . . . . . . . 16 SI3 ( Fns ⊗ ( S Image2nd )) V
106105ins2ex 5797 . . . . . . . . . . . . . . 15 Ins2 SI3 ( Fns ⊗ ( S Image2nd )) V
10799, 106symdifex 4108 . . . . . . . . . . . . . 14 ( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) V
108 1cex 4142 . . . . . . . . . . . . . 14 1c V
109107, 108imaex 4747 . . . . . . . . . . . . 13 (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) V
110109complex 4104 . . . . . . . . . . . 12 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) V
111110ins4ex 5799 . . . . . . . . . . 11 Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) V
112 enex 6031 . . . . . . . . . . . . 13 V
113112ins2ex 5797 . . . . . . . . . . . 12 Ins2 V
114113ins2ex 5797 . . . . . . . . . . 11 Ins2 Ins2 V
115111, 114inex 4105 . . . . . . . . . 10 ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) V
116115rnex 5107 . . . . . . . . 9 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) V
117116si3ex 5806 . . . . . . . 8 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) V
118117ins4ex 5799 . . . . . . 7 Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) V
11998, 118inex 4105 . . . . . 6 ( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) V
120108pw1ex 4303 . . . . . 6 11c V
121119, 120imaex 4747 . . . . 5 (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c) V
12295, 121inex 4105 . . . 4 ( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) V
123122, 120imaex 4747 . . 3 (( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) “ 11c) V
12489, 89, 123mpt2exlem 5811 . 2 ((( NC × NC ) × V) (( Ins2 S Ins3 (( Ins2 Ins3 ( S SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S Ins2 SI3 ( Fns ⊗ ( S Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ 11c)) “ 11c)) “ 1c)) V
12588, 124eqeltri 2423 1 c V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206   ∩ cin 3208   ⊕ csymdif 3209   ⊆ wss 3257  {csn 3737  1cc1c 4134  ℘1cpw1 4135  ⟨cop 4561   class class class wbr 4639   S csset 4719   SI csi 4720   ∘ ccom 4721   “ cima 4722   × cxp 4770  ran crn 4773   Fn wfn 4776  –→wf 4777  2nd c2nd 4783  (class class class)co 5525   ↦ cmpt2 5653   ⊗ 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   NC cncs 6088   ↑c cce 6096 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-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-ce 6106 This theorem is referenced by:  ce0nn  6180  spacvallem1  6281
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