Theorem ixpsnf1o 7948

Description: A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Hypothesis

Ref	Expression
ixpsnf1o.f	|- F = ( x e. A |-> ( { I } X. { x } ) )

Assertion

Ref	Expression
ixpsnf1o	|- ( I e. V -> F : A -1-1-onto-> X_ y e. { I } A )

Distinct variable groups:   x, I, y   x, A, y   x, V, y   y, F

Allowed substitution hint:   F( x)

Proof of Theorem ixpsnf1o

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ixpsnf1o.f	. 2 |- F = ( x e. A |-> ( { I } X. { x } ) )
2		snex 4908	. . . 4 |- { I } e. _V
3		snex 4908	. . . 4 |- { x } e. _V
4	2, 3	xpex 6962	. . 3 |- ( { I } X. { x } ) e. _V
5	4	a1i 11	. 2 |- ( ( I e. V /\ x e. A ) -> ( { I } X. { x } ) e. _V )
6		vex 3203	. . . . 5 |- a e. _V
7	6	rnex 7100	. . . 4 |- ran a e. _V
8	7	uniex 6953	. . 3 |- U. ran a e. _V
9	8	a1i 11	. 2 |- ( ( I e. V /\ a e. X_ y e. { I } A ) -> U. ran a e. _V )
10		sneq 4187	. . . . . 6 |- ( b = I -> { b } = { I } )
11	10	xpeq1d 5138	. . . . 5 |- ( b = I -> ( { b } X. { x } ) = ( { I } X. { x } ) )
12	11	eqeq2d 2632	. . . 4 |- ( b = I -> ( a = ( { b } X. { x } ) <-> a = ( { I } X. { x } ) ) )
13	12	anbi2d 740	. . 3 |- ( b = I -> ( ( x e. A /\ a = ( { b } X. { x } ) ) <-> ( x e. A /\ a = ( { I } X. { x } ) ) ) )
14		vex 3203	. . . . . 6 |- b e. _V
15		elixpsn 7947	. . . . . 6 |- ( b e. _V -> ( a e. X_ y e. { b } A <-> E. c e. A a = { <. b , c >. } ) )
16	14, 15	ax-mp 5	. . . . 5 |- ( a e. X_ y e. { b } A <-> E. c e. A a = { <. b , c >. } )
17	10	ixpeq1d 7920	. . . . . 6 |- ( b = I -> X_ y e. { b } A = X_ y e. { I } A )
18	17	eleq2d 2687	. . . . 5 |- ( b = I -> ( a e. X_ y e. { b } A <-> a e. X_ y e. { I } A ) )
19	16, 18	syl5bbr 274	. . . 4 |- ( b = I -> ( E. c e. A a = { <. b , c >. } <-> a e. X_ y e. { I } A ) )
20	19	anbi1d 741	. . 3 |- ( b = I -> ( ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) <-> ( a e. X_ y e. { I } A /\ x = U. ran a ) ) )
21		vex 3203	. . . . . . 7 |- x e. _V
22	14, 21	xpsn 6407	. . . . . 6 |- ( { b } X. { x } ) = { <. b , x >. }
23	22	eqeq2i 2634	. . . . 5 |- ( a = ( { b } X. { x } ) <-> a = { <. b , x >. } )
24	23	anbi2i 730	. . . 4 |- ( ( x e. A /\ a = ( { b } X. { x } ) ) <-> ( x e. A /\ a = { <. b , x >. } ) )
25		eqid 2622	. . . . . . . . 9 |- { <. b , x >. } = { <. b , x >. }
26		opeq2 4403	. . . . . . . . . . . 12 |- ( c = x -> <. b , c >. = <. b , x >. )
27	26	sneqd 4189	. . . . . . . . . . 11 |- ( c = x -> { <. b , c >. } = { <. b , x >. } )
28	27	eqeq2d 2632	. . . . . . . . . 10 |- ( c = x -> ( { <. b , x >. } = { <. b , c >. } <-> { <. b , x >. } = { <. b , x >. } ) )
29	28	rspcev 3309	. . . . . . . . 9 |- ( ( x e. A /\ { <. b , x >. } = { <. b , x >. } ) -> E. c e. A { <. b , x >. } = { <. b , c >. } )
30	25, 29	mpan2 707	. . . . . . . 8 |- ( x e. A -> E. c e. A { <. b , x >. } = { <. b , c >. } )
31	14, 21	op2nda 5620	. . . . . . . . 9 |- U. ran { <. b , x >. } = x
32	31	eqcomi 2631	. . . . . . . 8 |- x = U. ran { <. b , x >. }
33	30, 32	jctir 561	. . . . . . 7 |- ( x e. A -> ( E. c e. A { <. b , x >. } = { <. b , c >. } /\ x = U. ran { <. b , x >. } ) )
34		eqeq1 2626	. . . . . . . . 9 |- ( a = { <. b , x >. } -> ( a = { <. b , c >. } <-> { <. b , x >. } = { <. b , c >. } ) )
35	34	rexbidv 3052	. . . . . . . 8 |- ( a = { <. b , x >. } -> ( E. c e. A a = { <. b , c >. } <-> E. c e. A { <. b , x >. } = { <. b , c >. } ) )
36		rneq 5351	. . . . . . . . . 10 |- ( a = { <. b , x >. } -> ran a = ran { <. b , x >. } )
37	36	unieqd 4446	. . . . . . . . 9 |- ( a = { <. b , x >. } -> U. ran a = U. ran { <. b , x >. } )
38	37	eqeq2d 2632	. . . . . . . 8 |- ( a = { <. b , x >. } -> ( x = U. ran a <-> x = U. ran { <. b , x >. } ) )
39	35, 38	anbi12d 747	. . . . . . 7 |- ( a = { <. b , x >. } -> ( ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) <-> ( E. c e. A { <. b , x >. } = { <. b , c >. } /\ x = U. ran { <. b , x >. } ) ) )
40	33, 39	syl5ibrcom 237	. . . . . 6 |- ( x e. A -> ( a = { <. b , x >. } -> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) ) )
41	40	imp 445	. . . . 5 |- ( ( x e. A /\ a = { <. b , x >. } ) -> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) )
42		vex 3203	. . . . . . . . . . 11 |- c e. _V
43	14, 42	op2nda 5620	. . . . . . . . . 10 |- U. ran { <. b , c >. } = c
44	43	eqeq2i 2634	. . . . . . . . 9 |- ( x = U. ran { <. b , c >. } <-> x = c )
45		eqidd 2623	. . . . . . . . . . 11 |- ( c e. A -> { <. b , c >. } = { <. b , c >. } )
46	45	ancli 574	. . . . . . . . . 10 |- ( c e. A -> ( c e. A /\ { <. b , c >. } = { <. b , c >. } ) )
47		eleq1 2689	. . . . . . . . . . 11 |- ( x = c -> ( x e. A <-> c e. A ) )
48		opeq2 4403	. . . . . . . . . . . . 13 |- ( x = c -> <. b , x >. = <. b , c >. )
49	48	sneqd 4189	. . . . . . . . . . . 12 |- ( x = c -> { <. b , x >. } = { <. b , c >. } )
50	49	eqeq2d 2632	. . . . . . . . . . 11 |- ( x = c -> ( { <. b , c >. } = { <. b , x >. } <-> { <. b , c >. } = { <. b , c >. } ) )
51	47, 50	anbi12d 747	. . . . . . . . . 10 |- ( x = c -> ( ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) <-> ( c e. A /\ { <. b , c >. } = { <. b , c >. } ) ) )
52	46, 51	syl5ibrcom 237	. . . . . . . . 9 |- ( c e. A -> ( x = c -> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) )
53	44, 52	syl5bi 232	. . . . . . . 8 |- ( c e. A -> ( x = U. ran { <. b , c >. } -> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) )
54		rneq 5351	. . . . . . . . . . 11 |- ( a = { <. b , c >. } -> ran a = ran { <. b , c >. } )
55	54	unieqd 4446	. . . . . . . . . 10 |- ( a = { <. b , c >. } -> U. ran a = U. ran { <. b , c >. } )
56	55	eqeq2d 2632	. . . . . . . . 9 |- ( a = { <. b , c >. } -> ( x = U. ran a <-> x = U. ran { <. b , c >. } ) )
57		eqeq1 2626	. . . . . . . . . 10 |- ( a = { <. b , c >. } -> ( a = { <. b , x >. } <-> { <. b , c >. } = { <. b , x >. } ) )
58	57	anbi2d 740	. . . . . . . . 9 |- ( a = { <. b , c >. } -> ( ( x e. A /\ a = { <. b , x >. } ) <-> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) )
59	56, 58	imbi12d 334	. . . . . . . 8 |- ( a = { <. b , c >. } -> ( ( x = U. ran a -> ( x e. A /\ a = { <. b , x >. } ) ) <-> ( x = U. ran { <. b , c >. } -> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) ) )
60	53, 59	syl5ibrcom 237	. . . . . . 7 |- ( c e. A -> ( a = { <. b , c >. } -> ( x = U. ran a -> ( x e. A /\ a = { <. b , x >. } ) ) ) )
61	60	rexlimiv 3027	. . . . . 6 |- ( E. c e. A a = { <. b , c >. } -> ( x = U. ran a -> ( x e. A /\ a = { <. b , x >. } ) ) )
62	61	imp 445	. . . . 5 |- ( ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) -> ( x e. A /\ a = { <. b , x >. } ) )
63	41, 62	impbii 199	. . . 4 |- ( ( x e. A /\ a = { <. b , x >. } ) <-> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) )
64	24, 63	bitri 264	. . 3 |- ( ( x e. A /\ a = ( { b } X. { x } ) ) <-> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) )
65	13, 20, 64	vtoclbg 3267	. 2 |- ( I e. V -> ( ( x e. A /\ a = ( { I } X. { x } ) ) <-> ( a e. X_ y e. { I } A /\ x = U. ran a ) ) )
66	1, 5, 9, 65	f1od 6885	1 |- ( I e. V -> F : A -1-1-onto-> X_ y e. { I } A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   {csn 4177   <.cop 4183   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -1-1-onto->wf1o 5887   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ixp 7909

This theorem is referenced by:  mapsnf1o  7949