Theorem ceex 6175

Description: Cardinal exponentiation is stratified. (Contributed by SF, 3-Mar-2015.)

Assertion

Ref	Expression
ceex	|- ↑c e. _V

Proof of Theorem ceex

Dummy variables a b g m n x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ce 6107	. . 3 |- ↑c = (n e. NC , m e. NC |-> {g | E.aE.b(~P1 a e. n /\ ~P1 b e. m /\ g ~~ (a ^m b))})
2		snex 4112	. . . . . . . . . 10 |- {x} e. _V
3	2	otelins2 5792	. . . . . . . . 9 |- (<.{{a}}, <.{x}, <.n, m>.>.>. e. Ins2 Ins3 ( S o. SI Pw1Fn ) <-> <.{{a}}, <.n, m>.>. e. Ins3 ( S o. SI Pw1Fn ))
4		vex 2863	. . . . . . . . . 10 |- m e. _V
5	4	otelins3 5793	. . . . . . . . 9 |- (<.{{a}}, <.n, m>.>. e. Ins3 ( S o. SI Pw1Fn ) <-> <.{{a}}, n>. e. ( S o. SI Pw1Fn ))
6		ceexlem1 6174	. . . . . . . . 9 |- (<.{{a}}, n>. e. ( S o. SI Pw1Fn ) <-> ~P1 a e. n)
7	3, 5, 6	3bitri 262	. . . . . . . 8 |- (<.{{a}}, <.{x}, <.n, m>.>.>. e. Ins2 Ins3 ( S o. SI Pw1Fn ) <-> ~P1 a e. n)
8		elimapw11c 4949	. . . . . . . . 9 |- (<.{{a}}, <.{x}, <.n, m>.>.>. e. (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c) <-> E.b<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. ( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ )))
9		elin 3220	. . . . . . . . . . 11 |- (<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. ( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ )) <-> (<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) /\ <.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ )))
10		snex 4112	. . . . . . . . . . . . . 14 |- {{a}} e. _V
11	10	otelins2 5792	. . . . . . . . . . . . 13 |- (<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) <-> <.{{b}}, <.{x}, <.n, m>.>.>. e. Ins2 Ins2 ( S o. SI Pw1Fn ))
12	2	otelins2 5792	. . . . . . . . . . . . 13 |- (<.{{b}}, <.{x}, <.n, m>.>.>. e. Ins2 Ins2 ( S o. SI Pw1Fn ) <-> <.{{b}}, <.n, m>.>. e. Ins2 ( S o. SI Pw1Fn ))
13		vex 2863	. . . . . . . . . . . . . . 15 |- n e. _V
14	13	otelins2 5792	. . . . . . . . . . . . . 14 |- (<.{{b}}, <.n, m>.>. e. Ins2 ( S o. SI Pw1Fn ) <-> <.{{b}}, m>. e. ( S o. SI Pw1Fn ))
15		ceexlem1 6174	. . . . . . . . . . . . . 14 |- (<.{{b}}, m>. e. ( S o. SI Pw1Fn ) <-> ~P1 b e. m)
16	14, 15	bitri 240	. . . . . . . . . . . . 13 |- (<.{{b}}, <.n, m>.>. e. Ins2 ( S o. SI Pw1Fn ) <-> ~P1 b e. m)
17	11, 12, 16	3bitri 262	. . . . . . . . . . . 12 |- (<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) <-> ~P1 b e. m)
18	13, 4	opex 4589	. . . . . . . . . . . . . 14 |- <.n, m>. e. _V
19	18	oqelins4 5795	. . . . . . . . . . . . 13 |- (<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) <-> <.{{b}}, <.{{a}}, {x}>.>. e. SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))
20		snex 4112	. . . . . . . . . . . . . 14 |- {b} e. _V
21		snex 4112	. . . . . . . . . . . . . 14 |- {a} e. _V
22		vex 2863	. . . . . . . . . . . . . 14 |- x e. _V
23	20, 21, 22	otsnelsi3 5806	. . . . . . . . . . . . 13 |- (<.{{b}}, <.{{a}}, {x}>.>. e. SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) <-> <.{b}, <.{a}, x>.>. e. ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))
24		elrn2 4898	. . . . . . . . . . . . . 14 |- (<.{b}, <.{a}, x>.>. e. ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) <-> E.g<.g, <.{b}, <.{a}, x>.>.>. e. ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))
25		elin 3220	. . . . . . . . . . . . . . . 16 |- (<.g, <.{b}, <.{a}, x>.>.>. e. ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) <-> (<.g, <.{b}, <.{a}, x>.>.>. e. Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) /\ <.g, <.{b}, <.{a}, x>.>.>. e. Ins2 Ins2 ~~ ))
26	22	oqelins4 5795	. . . . . . . . . . . . . . . . . 18 |- (<.g, <.{b}, <.{a}, x>.>.>. e. Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) <-> <.g, <.{b}, {a}>.>. e. ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c))
27	20, 21	opex 4589	. . . . . . . . . . . . . . . . . . 19 |- <.{b}, {a}>. e. _V
28		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 |- f e. _V
29	28	brfns 5834	. . . . . . . . . . . . . . . . . . . . . 22 |- (f Fns b <-> f Fn b)
30		brco 4884	. . . . . . . . . . . . . . . . . . . . . . 23 |- (f( S o. Image2nd)a <-> E.x(fImage2ndx /\ x S a))
31	28, 22	brimage 5794	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (fImage2ndx <-> x = (2nd"f))
32		dfrn5 5509	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ran f = (2nd"f)
33	32	eqeq2i 2363	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x = ran f <-> x = (2nd"f))
34	31, 33	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (fImage2ndx <-> x = ran f)
35		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- a e. _V
36	22, 35	brsset 4759	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x S a <-> x C_ a)
37	34, 36	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((fImage2ndx /\ x S a) <-> (x = ran f /\ x C_ a))
38	37	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . 23 |- (E.x(fImage2ndx /\ x S a) <-> E.x(x = ran f /\ x C_ a))
39	28	rnex 5108	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ran f e. _V
40		sseq1 3293	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x = ran f -> (x C_ a <-> ran f C_ a))
41	39, 40	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . 23 |- (E.x(x = ran f /\ x C_ a) <-> ran f C_ a)
42	30, 38, 41	3bitri 262	. . . . . . . . . . . . . . . . . . . . . 22 |- (f( S o. Image2nd)a <-> ran f C_ a)
43	29, 42	anbi12i 678	. . . . . . . . . . . . . . . . . . . . 21 |- ((f Fns b /\ f( S o. Image2nd)a) <-> (f Fn b /\ ran f C_ a))
44		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 |- (f( Fns (x) ( S o. Image2nd))<.b, a>. <-> <.f, <.b, a>.>. e. ( Fns (x) ( S o. Image2nd)))
45		trtxp 5782	. . . . . . . . . . . . . . . . . . . . . 22 |- (f( Fns (x) ( S o. Image2nd))<.b, a>. <-> (f Fns b /\ f( S o. Image2nd)a))
46	44, 45	bitr3i 242	. . . . . . . . . . . . . . . . . . . . 21 |- (<.f, <.b, a>.>. e. ( Fns (x) ( S o. Image2nd)) <-> (f Fns b /\ f( S o. Image2nd)a))
47		df-f 4792	. . . . . . . . . . . . . . . . . . . . 21 |- (f:b-->a <-> (f Fn b /\ ran f C_ a))
48	43, 46, 47	3bitr4i 268	. . . . . . . . . . . . . . . . . . . 20 |- (<.f, <.b, a>.>. e. ( Fns (x) ( S o. Image2nd)) <-> f:b-->a)
49		vex 2863	. . . . . . . . . . . . . . . . . . . . 21 |- b e. _V
50	28, 49, 35	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . 20 |- (<.{f}, <.{b}, {a}>.>. e. SI3 ( Fns (x) ( S o. Image2nd)) <-> <.f, <.b, a>.>. e. ( Fns (x) ( S o. Image2nd)))
51	35, 49, 28	elmap 6018	. . . . . . . . . . . . . . . . . . . 20 |- (f e. (a ^m b) <-> f:b-->a)
52	48, 50, 51	3bitr4i 268	. . . . . . . . . . . . . . . . . . 19 |- (<.{f}, <.{b}, {a}>.>. e. SI3 ( Fns (x) ( S o. Image2nd)) <-> f e. (a ^m b))
53	27, 52	releqel 5808	. . . . . . . . . . . . . . . . . 18 |- (<.g, <.{b}, {a}>.>. e. ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) <-> g = (a ^m b))
54	26, 53	bitri 240	. . . . . . . . . . . . . . . . 17 |- (<.g, <.{b}, <.{a}, x>.>.>. e. Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) <-> g = (a ^m b))
55	20	otelins2 5792	. . . . . . . . . . . . . . . . . 18 |- (<.g, <.{b}, <.{a}, x>.>.>. e. Ins2 Ins2 ~~ <-> <.g, <.{a}, x>.>. e. Ins2 ~~ )
56	21	otelins2 5792	. . . . . . . . . . . . . . . . . . 19 |- (<.g, <.{a}, x>.>. e. Ins2 ~~ <-> <.g, x>. e. ~~ )
57		df-br 4641	. . . . . . . . . . . . . . . . . . 19 |- (g ~~ x <-> <.g, x>. e. ~~ )
58	56, 57	bitr4i 243	. . . . . . . . . . . . . . . . . 18 |- (<.g, <.{a}, x>.>. e. Ins2 ~~ <-> g ~~ x)
59		ensym 6038	. . . . . . . . . . . . . . . . . 18 |- (g ~~ x <-> x ~~ g)
60	55, 58, 59	3bitri 262	. . . . . . . . . . . . . . . . 17 |- (<.g, <.{b}, <.{a}, x>.>.>. e. Ins2 Ins2 ~~ <-> x ~~ g)
61	54, 60	anbi12i 678	. . . . . . . . . . . . . . . 16 |- ((<.g, <.{b}, <.{a}, x>.>.>. e. Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) /\ <.g, <.{b}, <.{a}, x>.>.>. e. Ins2 Ins2 ~~ ) <-> (g = (a ^m b) /\ x ~~ g))
62	25, 61	bitri 240	. . . . . . . . . . . . . . 15 |- (<.g, <.{b}, <.{a}, x>.>.>. e. ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) <-> (g = (a ^m b) /\ x ~~ g))
63	62	exbii 1582	. . . . . . . . . . . . . 14 |- (E.g<.g, <.{b}, <.{a}, x>.>.>. e. ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) <-> E.g(g = (a ^m b) /\ x ~~ g))
64		ovex 5552	. . . . . . . . . . . . . . 15 |- (a ^m b) e. _V
65		breq2 4644	. . . . . . . . . . . . . . 15 |- (g = (a ^m b) -> (x ~~ g <-> x ~~ (a ^m b)))
66	64, 65	ceqsexv 2895	. . . . . . . . . . . . . 14 |- (E.g(g = (a ^m b) /\ x ~~ g) <-> x ~~ (a ^m b))
67	24, 63, 66	3bitri 262	. . . . . . . . . . . . 13 |- (<.{b}, <.{a}, x>.>. e. ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) <-> x ~~ (a ^m b))
68	19, 23, 67	3bitri 262	. . . . . . . . . . . 12 |- (<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) <-> x ~~ (a ^m b))
69	17, 68	anbi12i 678	. . . . . . . . . . 11 |- ((<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) /\ <.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ )) <-> (~P1 b e. m /\ x ~~ (a ^m b)))
70	9, 69	bitri 240	. . . . . . . . . 10 |- (<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. ( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ )) <-> (~P1 b e. m /\ x ~~ (a ^m b)))
71	70	exbii 1582	. . . . . . . . 9 |- (E.b<.{{b}}, <.{{a}}, <.{x}, <.n, m>.>.>.>. e. ( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ )) <-> E.b(~P1 b e. m /\ x ~~ (a ^m b)))
72	8, 71	bitri 240	. . . . . . . 8 |- (<.{{a}}, <.{x}, <.n, m>.>.>. e. (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c) <-> E.b(~P1 b e. m /\ x ~~ (a ^m b)))
73	7, 72	anbi12i 678	. . . . . . 7 |- ((<.{{a}}, <.{x}, <.n, m>.>.>. e. Ins2 Ins3 ( S o. SI Pw1Fn ) /\ <.{{a}}, <.{x}, <.n, m>.>.>. e. (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c)) <-> (~P1 a e. n /\ E.b(~P1 b e. m /\ x ~~ (a ^m b))))
74		elin 3220	. . . . . . 7 |- (<.{{a}}, <.{x}, <.n, m>.>.>. e. ( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c)) <-> (<.{{a}}, <.{x}, <.n, m>.>.>. e. Ins2 Ins3 ( S o. SI Pw1Fn ) /\ <.{{a}}, <.{x}, <.n, m>.>.>. e. (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c)))
75		3anass 938	. . . . . . . . 9 |- ((~P1 a e. n /\ ~P1 b e. m /\ x ~~ (a ^m b)) <-> (~P1 a e. n /\ (~P1 b e. m /\ x ~~ (a ^m b))))
76	75	exbii 1582	. . . . . . . 8 |- (E.b(~P1 a e. n /\ ~P1 b e. m /\ x ~~ (a ^m b)) <-> E.b(~P1 a e. n /\ (~P1 b e. m /\ x ~~ (a ^m b))))
77		19.42v 1905	. . . . . . . 8 |- (E.b(~P1 a e. n /\ (~P1 b e. m /\ x ~~ (a ^m b))) <-> (~P1 a e. n /\ E.b(~P1 b e. m /\ x ~~ (a ^m b))))
78	76, 77	bitri 240	. . . . . . 7 |- (E.b(~P1 a e. n /\ ~P1 b e. m /\ x ~~ (a ^m b)) <-> (~P1 a e. n /\ E.b(~P1 b e. m /\ x ~~ (a ^m b))))
79	73, 74, 78	3bitr4i 268	. . . . . 6 |- (<.{{a}}, <.{x}, <.n, m>.>.>. e. ( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c)) <-> E.b(~P1 a e. n /\ ~P1 b e. m /\ x ~~ (a ^m b)))
80	79	exbii 1582	. . . . 5 |- (E.a<.{{a}}, <.{x}, <.n, m>.>.>. e. ( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c)) <-> E.aE.b(~P1 a e. n /\ ~P1 b e. m /\ x ~~ (a ^m b)))
81		elimapw11c 4949	. . . . 5 |- (<.{x}, <.n, m>.>. e. (( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c))"~P1 1c) <-> E.a<.{{a}}, <.{x}, <.n, m>.>.>. e. ( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c)))
82		breq1 4643	. . . . . . . 8 |- (g = x -> (g ~~ (a ^m b) <-> x ~~ (a ^m b)))
83	82	3anbi3d 1258	. . . . . . 7 |- (g = x -> ((~P1 a e. n /\ ~P1 b e. m /\ g ~~ (a ^m b)) <-> (~P1 a e. n /\ ~P1 b e. m /\ x ~~ (a ^m b))))
84	83	2exbidv 1628	. . . . . 6 |- (g = x -> (E.aE.b(~P1 a e. n /\ ~P1 b e. m /\ g ~~ (a ^m b)) <-> E.aE.b(~P1 a e. n /\ ~P1 b e. m /\ x ~~ (a ^m b))))
85	22, 84	elab 2986	. . . . 5 |- (x e. {g | E.aE.b(~P1 a e. n /\ ~P1 b e. m /\ g ~~ (a ^m b))} <-> E.aE.b(~P1 a e. n /\ ~P1 b e. m /\ x ~~ (a ^m b)))
86	80, 81, 85	3bitr4i 268	. . . 4 |- (<.{x}, <.n, m>.>. e. (( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c))"~P1 1c) <-> x e. {g | E.aE.b(~P1 a e. n /\ ~P1 b e. m /\ g ~~ (a ^m b))})
87	86	releqmpt2 5810	. . 3 |- ((( NC X. NC ) X. _V) \ (( Ins2 S (+) Ins3 (( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c))"~P1 1c))"1c)) = (n e. NC , m e. NC |-> {g | E.aE.b(~P1 a e. n /\ ~P1 b e. m /\ g ~~ (a ^m b))})
88	1, 87	eqtr4i 2376	. 2 |- ↑c = ((( NC X. NC ) X. _V) \ (( Ins2 S (+) Ins3 (( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c))"~P1 1c))"1c))
89		ncsex 6112	. . 3 |- NC e. _V
90		ssetex 4745	. . . . . . . 8 |- S e. _V
91		pw1fnex 5853	. . . . . . . . 9 |- Pw1Fn e. _V
92	91	siex 4754	. . . . . . . 8 |- SI Pw1Fn e. _V
93	90, 92	coex 4751	. . . . . . 7 |- ( S o. SI Pw1Fn ) e. _V
94	93	ins3ex 5799	. . . . . 6 |- Ins3 ( S o. SI Pw1Fn ) e. _V
95	94	ins2ex 5798	. . . . 5 |- Ins2 Ins3 ( S o. SI Pw1Fn ) e. _V
96	93	ins2ex 5798	. . . . . . . . 9 |- Ins2 ( S o. SI Pw1Fn ) e. _V
97	96	ins2ex 5798	. . . . . . . 8 |- Ins2 Ins2 ( S o. SI Pw1Fn ) e. _V
98	97	ins2ex 5798	. . . . . . 7 |- Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) e. _V
99	90	ins3ex 5799	. . . . . . . . . . . . . . 15 |- Ins3 S e. _V
100		fnsex 5833	. . . . . . . . . . . . . . . . . 18 |- Fns e. _V
101		2ndex 5113	. . . . . . . . . . . . . . . . . . . 20 |- 2nd e. _V
102	101	imageex 5802	. . . . . . . . . . . . . . . . . . 19 |- Image2nd e. _V
103	90, 102	coex 4751	. . . . . . . . . . . . . . . . . 18 |- ( S o. Image2nd) e. _V
104	100, 103	txpex 5786	. . . . . . . . . . . . . . . . 17 |- ( Fns (x) ( S o. Image2nd)) e. _V
105	104	si3ex 5807	. . . . . . . . . . . . . . . 16 |- SI3 ( Fns (x) ( S o. Image2nd)) e. _V
106	105	ins2ex 5798	. . . . . . . . . . . . . . 15 |- Ins2 SI3 ( Fns (x) ( S o. Image2nd)) e. _V
107	99, 106	symdifex 4109	. . . . . . . . . . . . . 14 |- ( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd))) e. _V
108		1cex 4143	. . . . . . . . . . . . . 14 |- 1c e. _V
109	107, 108	imaex 4748	. . . . . . . . . . . . 13 |- (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) e. _V
110	109	complex 4105	. . . . . . . . . . . 12 |- ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) e. _V
111	110	ins4ex 5800	. . . . . . . . . . 11 |- Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) e. _V
112		enex 6032	. . . . . . . . . . . . 13 |- ~~ e. _V
113	112	ins2ex 5798	. . . . . . . . . . . 12 |- Ins2 ~~ e. _V
114	113	ins2ex 5798	. . . . . . . . . . 11 |- Ins2 Ins2 ~~ e. _V
115	111, 114	inex 4106	. . . . . . . . . 10 |- ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) e. _V
116	115	rnex 5108	. . . . . . . . 9 |- ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) e. _V
117	116	si3ex 5807	. . . . . . . 8 |- SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) e. _V
118	117	ins4ex 5800	. . . . . . 7 |- Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ) e. _V
119	98, 118	inex 4106	. . . . . 6 |- ( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ )) e. _V
120	108	pw1ex 4304	. . . . . 6 |- ~P1 1c e. _V
121	119, 120	imaex 4748	. . . . 5 |- (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c) e. _V
122	95, 121	inex 4106	. . . 4 |- ( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c)) e. _V
123	122, 120	imaex 4748	. . 3 |- (( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c))"~P1 1c) e. _V
124	89, 89, 123	mpt2exlem 5812	. 2 |- ((( NC X. NC ) X. _V) \ (( Ins2 S (+) Ins3 (( Ins2 Ins3 ( S o. SI Pw1Fn ) i^i (( Ins2 Ins2 Ins2 ( S o. SI Pw1Fn ) i^i Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S (+) Ins2 SI3 ( Fns (x) ( S o. Image2nd)))"1c) i^i Ins2 Ins2 ~~ ))"~P1 1c))"~P1 1c))"1c)) e. _V
125	88, 124	eqeltri 2423	1 |- ↑c e. _V

Syntax hints:   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  _Vcvv 2860   ∼ ccompl 3206   \ cdif 3207   i^i cin 3209   (+) csymdif 3210   C_ wss 3258  {csn 3738  1_cc1c 4135  ~P₁ cpw1 4136  <.cop 4562   class class class wbr 4640   S csset 4720   SI csi 4721   o. ccom 4722  "cima 4723   X. cxp 4771  ran crn 4774   Fn wfn 4777  -->wf 4778  2ndc2nd 4784  (class class class)co 5526   |-> cmpt2 5654   (x) ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752  Imagecimage 5754   Ins4 cins4 5756   SI₃ csi3 5758   Fns cfns 5762   Pw1Fn cpw1fn 5766   ^m cmap 6000   ~~ cen 6029   NC cncs 6089   ↑_c cce 6097

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-ce 6107

This theorem is referenced by:  ce0nn  6181  spacvallem1  6282