Theorem ixpsnf1o 6714

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		snexg 4170	. . . 4 |- ( I e. V -> { I } e. _V )
3		vex 2733	. . . . 5 |- x e. _V
4	3	snex 4171	. . . 4 |- { x } e. _V
5		xpexg 4725	. . . 4 |- ( ( { I } e. _V /\ { x } e. _V ) -> ( { I } X. { x } ) e. _V )
6	2, 4, 5	sylancl 411	. . 3 |- ( I e. V -> ( { I } X. { x } ) e. _V )
7	6	adantr 274	. 2 |- ( ( I e. V /\ x e. A ) -> ( { I } X. { x } ) e. _V )
8		vex 2733	. . . . 5 |- a e. _V
9	8	rnex 4878	. . . 4 |- ran a e. _V
10	9	uniex 4422	. . 3 |- U. ran a e. _V
11	10	a1i 9	. 2 |- ( ( I e. V /\ a e. X_ y e. { I } A ) -> U. ran a e. _V )
12		sneq 3594	. . . . . 6 |- ( b = I -> { b } = { I } )
13	12	xpeq1d 4634	. . . . 5 |- ( b = I -> ( { b } X. { x } ) = ( { I } X. { x } ) )
14	13	eqeq2d 2182	. . . 4 |- ( b = I -> ( a = ( { b } X. { x } ) <-> a = ( { I } X. { x } ) ) )
15	14	anbi2d 461	. . 3 |- ( b = I -> ( ( x e. A /\ a = ( { b } X. { x } ) ) <-> ( x e. A /\ a = ( { I } X. { x } ) ) ) )
16		elixpsn 6713	. . . . . 6 |- ( b e. _V -> ( a e. X_ y e. { b } A <-> E. c e. A a = { <. b , c >. } ) )
17	16	elv 2734	. . . . 5 |- ( a e. X_ y e. { b } A <-> E. c e. A a = { <. b , c >. } )
18	12	ixpeq1d 6688	. . . . . 6 |- ( b = I -> X_ y e. { b } A = X_ y e. { I } A )
19	18	eleq2d 2240	. . . . 5 |- ( b = I -> ( a e. X_ y e. { b } A <-> a e. X_ y e. { I } A ) )
20	17, 19	bitr3id 193	. . . 4 |- ( b = I -> ( E. c e. A a = { <. b , c >. } <-> a e. X_ y e. { I } A ) )
21	20	anbi1d 462	. . 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 ) ) )
22		vex 2733	. . . . . . 7 |- b e. _V
23	22, 3	xpsn 5672	. . . . . 6 |- ( { b } X. { x } ) = { <. b , x >. }
24	23	eqeq2i 2181	. . . . 5 |- ( a = ( { b } X. { x } ) <-> a = { <. b , x >. } )
25	24	anbi2i 454	. . . 4 |- ( ( x e. A /\ a = ( { b } X. { x } ) ) <-> ( x e. A /\ a = { <. b , x >. } ) )
26		eqid 2170	. . . . . . . . 9 |- { <. b , x >. } = { <. b , x >. }
27		opeq2 3766	. . . . . . . . . . 11 |- ( c = x -> <. b , c >. = <. b , x >. )
28	27	sneqd 3596	. . . . . . . . . 10 |- ( c = x -> { <. b , c >. } = { <. b , x >. } )
29	28	rspceeqv 2852	. . . . . . . . 9 |- ( ( x e. A /\ { <. b , x >. } = { <. b , x >. } ) -> E. c e. A { <. b , x >. } = { <. b , c >. } )
30	26, 29	mpan2 423	. . . . . . . 8 |- ( x e. A -> E. c e. A { <. b , x >. } = { <. b , c >. } )
31	22, 3	op2nda 5095	. . . . . . . . 9 |- U. ran { <. b , x >. } = x
32	31	eqcomi 2174	. . . . . . . 8 |- x = U. ran { <. b , x >. }
33	30, 32	jctir 311	. . . . . . 7 |- ( x e. A -> ( E. c e. A { <. b , x >. } = { <. b , c >. } /\ x = U. ran { <. b , x >. } ) )
34		eqeq1 2177	. . . . . . . . 9 |- ( a = { <. b , x >. } -> ( a = { <. b , c >. } <-> { <. b , x >. } = { <. b , c >. } ) )
35	34	rexbidv 2471	. . . . . . . 8 |- ( a = { <. b , x >. } -> ( E. c e. A a = { <. b , c >. } <-> E. c e. A { <. b , x >. } = { <. b , c >. } ) )
36		rneq 4838	. . . . . . . . . 10 |- ( a = { <. b , x >. } -> ran a = ran { <. b , x >. } )
37	36	unieqd 3807	. . . . . . . . 9 |- ( a = { <. b , x >. } -> U. ran a = U. ran { <. b , x >. } )
38	37	eqeq2d 2182	. . . . . . . 8 |- ( a = { <. b , x >. } -> ( x = U. ran a <-> x = U. ran { <. b , x >. } ) )
39	35, 38	anbi12d 470	. . . . . . 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 156	. . . . . 6 |- ( x e. A -> ( a = { <. b , x >. } -> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) ) )
41	40	imp 123	. . . . 5 |- ( ( x e. A /\ a = { <. b , x >. } ) -> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) )
42		vex 2733	. . . . . . . . . . 11 |- c e. _V
43	22, 42	op2nda 5095	. . . . . . . . . 10 |- U. ran { <. b , c >. } = c
44	43	eqeq2i 2181	. . . . . . . . 9 |- ( x = U. ran { <. b , c >. } <-> x = c )
45		eqidd 2171	. . . . . . . . . . 11 |- ( c e. A -> { <. b , c >. } = { <. b , c >. } )
46	45	ancli 321	. . . . . . . . . 10 |- ( c e. A -> ( c e. A /\ { <. b , c >. } = { <. b , c >. } ) )
47		eleq1w 2231	. . . . . . . . . . 11 |- ( x = c -> ( x e. A <-> c e. A ) )
48		opeq2 3766	. . . . . . . . . . . . 13 |- ( x = c -> <. b , x >. = <. b , c >. )
49	48	sneqd 3596	. . . . . . . . . . . 12 |- ( x = c -> { <. b , x >. } = { <. b , c >. } )
50	49	eqeq2d 2182	. . . . . . . . . . 11 |- ( x = c -> ( { <. b , c >. } = { <. b , x >. } <-> { <. b , c >. } = { <. b , c >. } ) )
51	47, 50	anbi12d 470	. . . . . . . . . 10 |- ( x = c -> ( ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) <-> ( c e. A /\ { <. b , c >. } = { <. b , c >. } ) ) )
52	46, 51	syl5ibrcom 156	. . . . . . . . 9 |- ( c e. A -> ( x = c -> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) )
53	44, 52	syl5bi 151	. . . . . . . 8 |- ( c e. A -> ( x = U. ran { <. b , c >. } -> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) )
54		rneq 4838	. . . . . . . . . . 11 |- ( a = { <. b , c >. } -> ran a = ran { <. b , c >. } )
55	54	unieqd 3807	. . . . . . . . . 10 |- ( a = { <. b , c >. } -> U. ran a = U. ran { <. b , c >. } )
56	55	eqeq2d 2182	. . . . . . . . 9 |- ( a = { <. b , c >. } -> ( x = U. ran a <-> x = U. ran { <. b , c >. } ) )
57		eqeq1 2177	. . . . . . . . . 10 |- ( a = { <. b , c >. } -> ( a = { <. b , x >. } <-> { <. b , c >. } = { <. b , x >. } ) )
58	57	anbi2d 461	. . . . . . . . 9 |- ( a = { <. b , c >. } -> ( ( x e. A /\ a = { <. b , x >. } ) <-> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) )
59	56, 58	imbi12d 233	. . . . . . . 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 156	. . . . . . 7 |- ( c e. A -> ( a = { <. b , c >. } -> ( x = U. ran a -> ( x e. A /\ a = { <. b , x >. } ) ) ) )
61	60	rexlimiv 2581	. . . . . 6 |- ( E. c e. A a = { <. b , c >. } -> ( x = U. ran a -> ( x e. A /\ a = { <. b , x >. } ) ) )
62	61	imp 123	. . . . 5 |- ( ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) -> ( x e. A /\ a = { <. b , x >. } ) )
63	41, 62	impbii 125	. . . 4 |- ( ( x e. A /\ a = { <. b , x >. } ) <-> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) )
64	25, 63	bitri 183	. . 3 |- ( ( x e. A /\ a = ( { b } X. { x } ) ) <-> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) )
65	15, 21, 64	vtoclbg 2791	. 2 |- ( I e. V -> ( ( x e. A /\ a = ( { I } X. { x } ) ) <-> ( a e. X_ y e. { I } A /\ x = U. ran a ) ) )
66	1, 7, 11, 65	f1od 6052	1 |- ( I e. V -> F : A -1-1-onto-> X_ y e. { I } A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449   _Vcvv 2730   {csn 3583   <.cop 3586   U.cuni 3796   |-> cmpt 4050   X. cxp 4609   ran crn 4612   -1-1-onto->wf1o 5197   X_cixp 6676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ixp 6677

This theorem is referenced by:  mapsnf1o  6715