Theorem ixpsnf1o 6900

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 4272	. . . 4 |- ( I e. V -> { I } e. _V )
3		vex 2803	. . . . 5 |- x e. _V
4	3	snex 4273	. . . 4 |- { x } e. _V
5		xpexg 4838	. . . 4 |- ( ( { I } e. _V /\ { x } e. _V ) -> ( { I } X. { x } ) e. _V )
6	2, 4, 5	sylancl 413	. . 3 |- ( I e. V -> ( { I } X. { x } ) e. _V )
7	6	adantr 276	. 2 |- ( ( I e. V /\ x e. A ) -> ( { I } X. { x } ) e. _V )
8		vex 2803	. . . . 5 |- a e. _V
9	8	rnex 4998	. . . 4 |- ran a e. _V
10	9	uniex 4532	. . 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 3678	. . . . . 6 |- ( b = I -> { b } = { I } )
13	12	xpeq1d 4746	. . . . 5 |- ( b = I -> ( { b } X. { x } ) = ( { I } X. { x } ) )
14	13	eqeq2d 2241	. . . 4 |- ( b = I -> ( a = ( { b } X. { x } ) <-> a = ( { I } X. { x } ) ) )
15	14	anbi2d 464	. . 3 |- ( b = I -> ( ( x e. A /\ a = ( { b } X. { x } ) ) <-> ( x e. A /\ a = ( { I } X. { x } ) ) ) )
16		elixpsn 6899	. . . . . 6 |- ( b e. _V -> ( a e. X_ y e. { b } A <-> E. c e. A a = { <. b , c >. } ) )
17	16	elv 2804	. . . . 5 |- ( a e. X_ y e. { b } A <-> E. c e. A a = { <. b , c >. } )
18	12	ixpeq1d 6874	. . . . . 6 |- ( b = I -> X_ y e. { b } A = X_ y e. { I } A )
19	18	eleq2d 2299	. . . . 5 |- ( b = I -> ( a e. X_ y e. { b } A <-> a e. X_ y e. { I } A ) )
20	17, 19	bitr3id 194	. . . 4 |- ( b = I -> ( E. c e. A a = { <. b , c >. } <-> a e. X_ y e. { I } A ) )
21	20	anbi1d 465	. . 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 2803	. . . . . . 7 |- b e. _V
23	22, 3	xpsn 5819	. . . . . 6 |- ( { b } X. { x } ) = { <. b , x >. }
24	23	eqeq2i 2240	. . . . 5 |- ( a = ( { b } X. { x } ) <-> a = { <. b , x >. } )
25	24	anbi2i 457	. . . 4 |- ( ( x e. A /\ a = ( { b } X. { x } ) ) <-> ( x e. A /\ a = { <. b , x >. } ) )
26		eqid 2229	. . . . . . . . 9 |- { <. b , x >. } = { <. b , x >. }
27		opeq2 3861	. . . . . . . . . . 11 |- ( c = x -> <. b , c >. = <. b , x >. )
28	27	sneqd 3680	. . . . . . . . . 10 |- ( c = x -> { <. b , c >. } = { <. b , x >. } )
29	28	rspceeqv 2926	. . . . . . . . 9 |- ( ( x e. A /\ { <. b , x >. } = { <. b , x >. } ) -> E. c e. A { <. b , x >. } = { <. b , c >. } )
30	26, 29	mpan2 425	. . . . . . . 8 |- ( x e. A -> E. c e. A { <. b , x >. } = { <. b , c >. } )
31	22, 3	op2nda 5219	. . . . . . . . 9 |- U. ran { <. b , x >. } = x
32	31	eqcomi 2233	. . . . . . . 8 |- x = U. ran { <. b , x >. }
33	30, 32	jctir 313	. . . . . . 7 |- ( x e. A -> ( E. c e. A { <. b , x >. } = { <. b , c >. } /\ x = U. ran { <. b , x >. } ) )
34		eqeq1 2236	. . . . . . . . 9 |- ( a = { <. b , x >. } -> ( a = { <. b , c >. } <-> { <. b , x >. } = { <. b , c >. } ) )
35	34	rexbidv 2531	. . . . . . . 8 |- ( a = { <. b , x >. } -> ( E. c e. A a = { <. b , c >. } <-> E. c e. A { <. b , x >. } = { <. b , c >. } ) )
36		rneq 4957	. . . . . . . . . 10 |- ( a = { <. b , x >. } -> ran a = ran { <. b , x >. } )
37	36	unieqd 3902	. . . . . . . . 9 |- ( a = { <. b , x >. } -> U. ran a = U. ran { <. b , x >. } )
38	37	eqeq2d 2241	. . . . . . . 8 |- ( a = { <. b , x >. } -> ( x = U. ran a <-> x = U. ran { <. b , x >. } ) )
39	35, 38	anbi12d 473	. . . . . . 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 157	. . . . . 6 |- ( x e. A -> ( a = { <. b , x >. } -> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) ) )
41	40	imp 124	. . . . 5 |- ( ( x e. A /\ a = { <. b , x >. } ) -> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) )
42		vex 2803	. . . . . . . . . . 11 |- c e. _V
43	22, 42	op2nda 5219	. . . . . . . . . 10 |- U. ran { <. b , c >. } = c
44	43	eqeq2i 2240	. . . . . . . . 9 |- ( x = U. ran { <. b , c >. } <-> x = c )
45		eqidd 2230	. . . . . . . . . . 11 |- ( c e. A -> { <. b , c >. } = { <. b , c >. } )
46	45	ancli 323	. . . . . . . . . 10 |- ( c e. A -> ( c e. A /\ { <. b , c >. } = { <. b , c >. } ) )
47		eleq1w 2290	. . . . . . . . . . 11 |- ( x = c -> ( x e. A <-> c e. A ) )
48		opeq2 3861	. . . . . . . . . . . . 13 |- ( x = c -> <. b , x >. = <. b , c >. )
49	48	sneqd 3680	. . . . . . . . . . . 12 |- ( x = c -> { <. b , x >. } = { <. b , c >. } )
50	49	eqeq2d 2241	. . . . . . . . . . 11 |- ( x = c -> ( { <. b , c >. } = { <. b , x >. } <-> { <. b , c >. } = { <. b , c >. } ) )
51	47, 50	anbi12d 473	. . . . . . . . . 10 |- ( x = c -> ( ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) <-> ( c e. A /\ { <. b , c >. } = { <. b , c >. } ) ) )
52	46, 51	syl5ibrcom 157	. . . . . . . . 9 |- ( c e. A -> ( x = c -> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) )
53	44, 52	biimtrid 152	. . . . . . . 8 |- ( c e. A -> ( x = U. ran { <. b , c >. } -> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) )
54		rneq 4957	. . . . . . . . . . 11 |- ( a = { <. b , c >. } -> ran a = ran { <. b , c >. } )
55	54	unieqd 3902	. . . . . . . . . 10 |- ( a = { <. b , c >. } -> U. ran a = U. ran { <. b , c >. } )
56	55	eqeq2d 2241	. . . . . . . . 9 |- ( a = { <. b , c >. } -> ( x = U. ran a <-> x = U. ran { <. b , c >. } ) )
57		eqeq1 2236	. . . . . . . . . 10 |- ( a = { <. b , c >. } -> ( a = { <. b , x >. } <-> { <. b , c >. } = { <. b , x >. } ) )
58	57	anbi2d 464	. . . . . . . . 9 |- ( a = { <. b , c >. } -> ( ( x e. A /\ a = { <. b , x >. } ) <-> ( x e. A /\ { <. b , c >. } = { <. b , x >. } ) ) )
59	56, 58	imbi12d 234	. . . . . . . 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 157	. . . . . . 7 |- ( c e. A -> ( a = { <. b , c >. } -> ( x = U. ran a -> ( x e. A /\ a = { <. b , x >. } ) ) ) )
61	60	rexlimiv 2642	. . . . . 6 |- ( E. c e. A a = { <. b , c >. } -> ( x = U. ran a -> ( x e. A /\ a = { <. b , x >. } ) ) )
62	61	imp 124	. . . . 5 |- ( ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) -> ( x e. A /\ a = { <. b , x >. } ) )
63	41, 62	impbii 126	. . . 4 |- ( ( x e. A /\ a = { <. b , x >. } ) <-> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) )
64	25, 63	bitri 184	. . 3 |- ( ( x e. A /\ a = ( { b } X. { x } ) ) <-> ( E. c e. A a = { <. b , c >. } /\ x = U. ran a ) )
65	15, 21, 64	vtoclbg 2863	. 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 6221	1 |- ( I e. V -> F : A -1-1-onto-> X_ y e. { I } A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2800   {csn 3667   <.cop 3670   U.cuni 3891   |-> cmpt 4148   X. cxp 4721   ran crn 4724   -1-1-onto->wf1o 5323   X_cixp 6862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ixp 6863

This theorem is referenced by:  mapsnf1o  6901