Theorem ssenen 4493

Description: Equinumerosity of equinumerous subsets of a set.

Hypotheses

Ref	Expression
ssenen.1	|- A e. V
ssenen.2	|- B e. V

Assertion

Ref	Expression
ssenen	|- (A ~~ B -> {x | (x (_ A /\ x ~~ C)} ~~ {x | (x (_ B /\ x ~~ C)})

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem ssenen

Step	Hyp	Ref	Expression
1		ssenen.2	. . . 4 |- B e. V
2	1	bren 4368	. . 3 |- (A ~~ B <-> E.f f:A-1-1-onto->B)
3		ssenen.1	. . . . . . . 8 |- A e. V
4	3	pwex 2741	. . . . . . 7 |- P~A e. V
5	4	inex1 2712	. . . . . 6 |- (P~A i^i {x | x ~~ C}) e. V
6	5	a1i 8	. . . . 5 |- (f:A-1-1-onto->B -> (P~A i^i {x | x ~~ C}) e. V)
7		entrt 4404	. . . . . . . . . 10 |- (((f"y) ~~ y /\ y ~~ C) -> (f"y) ~~ C)
8		visset 1810	. . . . . . . . . . . 12 |- y e. V
9	8	f1imaen 4412	. . . . . . . . . . 11 |- ((f:A-1-1->B /\ y (_ A) -> (f"y) ~~ y)
10		f1of1 3683	. . . . . . . . . . 11 |- (f:A-1-1-onto->B -> f:A-1-1->B)
11	9, 10	sylan 448	. . . . . . . . . 10 |- ((f:A-1-1-onto->B /\ y (_ A) -> (f"y) ~~ y)
12	7, 11	sylan 448	. . . . . . . . 9 |- (((f:A-1-1-onto->B /\ y (_ A) /\ y ~~ C) -> (f"y) ~~ C)
13	12	exp31 376	. . . . . . . 8 |- (f:A-1-1-onto->B -> (y (_ A -> (y ~~ C -> (f"y) ~~ C)))
14	13	imp3a 361	. . . . . . 7 |- (f:A-1-1-onto->B -> ((y (_ A /\ y ~~ C) -> (f"y) ~~ C))
15		f1ofo 3690	. . . . . . . 8 |- (f:A-1-1-onto->B -> f:A-onto->B)
16		imassrn 3411	. . . . . . . . 9 |- (f"y) (_ ran f
17		forn 3669	. . . . . . . . . 10 |- (f:A-onto->B -> ran f = B)
18	17	sseq2d 2086	. . . . . . . . 9 |- (f:A-onto->B -> ((f"y) (_ ran f <-> (f"y) (_ B))
19	16, 18	mpbii 193	. . . . . . . 8 |- (f:A-onto->B -> (f"y) (_ B)
20	15, 19	syl 10	. . . . . . 7 |- (f:A-1-1-onto->B -> (f"y) (_ B)
21	14, 20	jctild 600	. . . . . 6 |- (f:A-1-1-onto->B -> ((y (_ A /\ y ~~ C) -> ((f"y) (_ B /\ (f"y) ~~ C)))
22		elin 2204	. . . . . . 7 |- (y e. (P~A i^i {x | x ~~ C}) <-> (y e. P~A /\ y e. {x | x ~~ C}))
23	8	elpw 2401	. . . . . . . 8 |- (y e. P~A <-> y (_ A)
24		breq1 2618	. . . . . . . . 9 |- (x = y -> (x ~~ C <-> y ~~ C))
25	8, 24	elab 1894	. . . . . . . 8 |- (y e. {x | x ~~ C} <-> y ~~ C)
26	23, 25	anbi12i 482	. . . . . . 7 |- ((y e. P~A /\ y e. {x | x ~~ C}) <-> (y (_ A /\ y ~~ C))
27	22, 26	bitr 173	. . . . . 6 |- (y e. (P~A i^i {x | x ~~ C}) <-> (y (_ A /\ y ~~ C))
28		elin 2204	. . . . . . 7 |- ((f"y) e. (P~B i^i {x | x ~~ C}) <-> ((f"y) e. P~B /\ (f"y) e. {x | x ~~ C}))
29		visset 1810	. . . . . . . . . 10 |- f e. V
30		imaexg 3412	. . . . . . . . . 10 |- (f e. V -> (f"y) e. V)
31	29, 30	ax-mp 7	. . . . . . . . 9 |- (f"y) e. V
32	31	elpw 2401	. . . . . . . 8 |- ((f"y) e. P~B <-> (f"y) (_ B)
33		breq1 2618	. . . . . . . . 9 |- (x = (f"y) -> (x ~~ C <-> (f"y) ~~ C))
34	31, 33	elab 1894	. . . . . . . 8 |- ((f"y) e. {x | x ~~ C} <-> (f"y) ~~ C)
35	32, 34	anbi12i 482	. . . . . . 7 |- (((f"y) e. P~B /\ (f"y) e. {x | x ~~ C}) <-> ((f"y) (_ B /\ (f"y) ~~ C))
36	28, 35	bitr 173	. . . . . 6 |- ((f"y) e. (P~B i^i {x | x ~~ C}) <-> ((f"y) (_ B /\ (f"y) ~~ C))
37	21, 27, 36	3imtr4g 552	. . . . 5 |- (f:A-1-1-onto->B -> (y e. (P~A i^i {x | x ~~ C}) -> (f"y) e. (P~B i^i {x | x ~~ C})))
38		f1ocnv 3696	. . . . . . 7 |- (f:A-1-1-onto->B -> `'f:B-1-1-onto->A)
39		entrt 4404	. . . . . . . . . . 11 |- (((`'f"z) ~~ z /\ z ~~ C) -> (`'f"z) ~~ C)
40		visset 1810	. . . . . . . . . . . . 13 |- z e. V
41	40	f1imaen 4412	. . . . . . . . . . . 12 |- ((`'f:B-1-1->A /\ z (_ B) -> (`'f"z) ~~ z)
42		f1of1 3683	. . . . . . . . . . . 12 |- (`'f:B-1-1-onto->A -> `'f:B-1-1->A)
43	41, 42	sylan 448	. . . . . . . . . . 11 |- ((`'f:B-1-1-onto->A /\ z (_ B) -> (`'f"z) ~~ z)
44	39, 43	sylan 448	. . . . . . . . . 10 |- (((`'f:B-1-1-onto->A /\ z (_ B) /\ z ~~ C) -> (`'f"z) ~~ C)
45	44	exp31 376	. . . . . . . . 9 |- (`'f:B-1-1-onto->A -> (z (_ B -> (z ~~ C -> (`'f"z) ~~ C)))
46	45	imp3a 361	. . . . . . . 8 |- (`'f:B-1-1-onto->A -> ((z (_ B /\ z ~~ C) -> (`'f"z) ~~ C))
47		f1ofo 3690	. . . . . . . . 9 |- (`'f:B-1-1-onto->A -> `'f:B-onto->A)
48		imassrn 3411	. . . . . . . . . 10 |- (`'f"z) (_ ran `' f
49		forn 3669	. . . . . . . . . . 11 |- (`'f:B-onto->A -> ran `' f = A)
50	49	sseq2d 2086	. . . . . . . . . 10 |- (`'f:B-onto->A -> ((`'f"z) (_ ran `' f <-> (`'f"z) (_ A))
51	48, 50	mpbii 193	. . . . . . . . 9 |- (`'f:B-onto->A -> (`'f"z) (_ A)
52	47, 51	syl 10	. . . . . . . 8 |- (`'f:B-1-1-onto->A -> (`'f"z) (_ A)
53	46, 52	jctild 600	. . . . . . 7 |- (`'f:B-1-1-onto->A -> ((z (_ B /\ z ~~ C) -> ((`'f"z) (_ A /\ (`'f"z) ~~ C)))
54	38, 53	syl 10	. . . . . 6 |- (f:A-1-1-onto->B -> ((z (_ B /\ z ~~ C) -> ((`'f"z) (_ A /\ (`'f"z) ~~ C)))
55		elin 2204	. . . . . . 7 |- (z e. (P~B i^i {x | x ~~ C}) <-> (z e. P~B /\ z e. {x | x ~~ C}))
56	40	elpw 2401	. . . . . . . 8 |- (z e. P~B <-> z (_ B)
57		breq1 2618	. . . . . . . . 9 |- (x = z -> (x ~~ C <-> z ~~ C))
58	40, 57	elab 1894	. . . . . . . 8 |- (z e. {x | x ~~ C} <-> z ~~ C)
59	56, 58	anbi12i 482	. . . . . . 7 |- ((z e. P~B /\ z e. {x | x ~~ C}) <-> (z (_ B /\ z ~~ C))
60	55, 59	bitr 173	. . . . . 6 |- (z e. (P~B i^i {x | x ~~ C}) <-> (z (_ B /\ z ~~ C))
61		elin 2204	. . . . . . 7 |- ((`'f"z) e. (P~A i^i {x | x ~~ C}) <-> ((`'f"z) e. P~A /\ (`'f"z) e. {x | x ~~ C}))
62	29	cnvex 3516	. . . . . . . . . 10 |- `'f e. V
63		imaexg 3412	. . . . . . . . . 10 |- (`'f e. V -> (`'f"z) e. V)
64	62, 63	ax-mp 7	. . . . . . . . 9 |- (`'f"z) e. V
65	64	elpw 2401	. . . . . . . 8 |- ((`'f"z) e. P~A <-> (`'f"z) (_ A)
66		breq1 2618	. . . . . . . . 9 |- (x = (`'f"z) -> (x ~~ C <-> (`'f"z) ~~ C))
67	64, 66	elab 1894	. . . . . . . 8 |- ((`'f"z) e. {x | x ~~ C} <-> (`'f"z) ~~ C)
68	65, 67	anbi12i 482	. . . . . . 7 |- (((`'f"z) e. P~A /\ (`'f"z) e. {x | x ~~ C}) <-> ((`'f"z) (_ A /\ (`'f"z) ~~ C))
69	61, 68	bitr 173	. . . . . 6 |- ((`'f"z) e. (P~A i^i {x | x ~~ C}) <-> ((`'f"z) (_ A /\ (`'f"z) ~~ C))
70	54, 60, 69	3imtr4g 552	. . . . 5 |- (f:A-1-1-onto->B -> (z e. (P~B i^i {x | x ~~ C}) -> (`'f"z) e. (P~A i^i {x | x ~~ C})))
71		imaeq2 3398	. . . . . . . . . . . 12 |- (y = (`'f"z) -> (f"y) = (f"(`'f"z)))
72		f1orel 3687	. . . . . . . . . . . . . . . 16 |- (f:A-1-1-onto->B -> Rel f)
73		dfrel2 3481	. . . . . . . . . . . . . . . 16 |- (Rel f <-> `'`'f = f)
74	72, 73	sylib 198	. . . . . . . . . . . . . . 15 |- (f:A-1-1-onto->B -> `'`'f = f)
75	74	imaeq1d 3399	. . . . . . . . . . . . . 14 |- (f:A-1-1-onto->B -> (`'`'f"(`'f"z)) = (f"(`'f"z)))
76	75	adantr 389	. . . . . . . . . . . . 13 |- ((f:A-1-1-onto->B /\ z (_ B) -> (`'`'f"(`'f"z)) = (f"(`'f"z)))
77		f1imacnv 3700	. . . . . . . . . . . . . 14 |- ((`'f:B-1-1->A /\ z (_ B) -> (`'`'f"(`'f"z)) = z)
78	38, 42	syl 10	. . . . . . . . . . . . . 14 |- (f:A-1-1-onto->B -> `'f:B-1-1->A)
79	77, 78	sylan 448	. . . . . . . . . . . . 13 |- ((f:A-1-1-onto->B /\ z (_ B) -> (`'`'f"(`'f"z)) = z)
80	76, 79	eqtr3d 1507	. . . . . . . . . . . 12 |- ((f:A-1-1-onto->B /\ z (_ B) -> (f"(`'f"z)) = z)
81	71, 80	sylan9eqr 1527	. . . . . . . . . . 11 |- (((f:A-1-1-onto->B /\ z (_ B) /\ y = (`'f