Theorem ixpsnf1o 6904

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 4274	. . . 4 |- ( I e. V -> { I } e. _V )
3		vex 2805	. . . . 5 |- x e. _V
4	3	snex 4275	. . . 4 |- { x } e. _V
5		xpexg 4840	. . . 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 2805	. . . . 5 |- a e. _V
9	8	rnex 5000	. . . 4 |- ran a e. _V
10	9	uniex 4534	. . 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 3680	. . . . . 6 |- ( b = I -> { b } = { I } )
13	12	xpeq1d 4748	. . . . 5 |- ( b = I -> ( { b } X. { x } ) = ( { I } X. { x } ) )
14	13	eqeq2d 2243	. . . 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 6903	. . . . . 6 |- ( b e. _V -> ( a e. X_ y e. { b } A <-> E. c e. A a = { <. b , c >. } ) )
17	16	elv 2806	. . . . 5 |- ( a e. X_ y e. { b } A <-> E. c e. A a = { <. b , c >. } )
18	12	ixpeq1d 6878	. . . . . 6 |- ( b = I -> X_ y e. { b } A = X_ y e. { I } A )
19	18	eleq2d 2301	. . . . 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 2805	. . . . . . 7 |- b e. _V
23	22, 3	xpsn 5823	. . . . . 6 |- ( { b } X. { x } ) = { <. b , x >. }
24	23	eqeq2i 2242	. . . . 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 2231	. . . . . . . . 9 |- { <. b , x >. } = { <. b , x >. }
27		opeq2 3863	. . . . . . . . . . 11 |- ( c = x -> <. b , c >. = <. b , x >. )
28	27	sneqd 3682	. . . . . . . . . 10 |- ( c = x -> { <. b , c >. } = { <. b , x >. } )
29	28	rspceeqv 2928	. . . . . . . . 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 5221	. . . . . . . . 9 |- U. ran { <. b , x >. } = x
32	31	eqcomi 2235	. . . . . . . 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 2238	. . . . . . . . 9 |- ( a = { <. b , x >. } -> ( a = { <. b , c >. } <-> { <. b , x >. } = { <. b , c >. } ) )
35	34	rexbidv 2533	. . . . . . . 8 |- ( a = { <. b , x >. } -> ( E. c e. A a = { <. b , c >. } <-> E. c e. A { <. b , x >. } = { <. b , c >. } ) )
36		rneq 4959	. . . . . . . . . 10 |- ( a = { <. b , x >. } -> ran a = ran { <. b , x >. } )
37	36	unieqd 3904	. . . . . . . . 9 |- ( a = { <. b , x >. } -> U. ran a = U. ran { <. b , x >. } )
38	37	eqeq2d 2243	. . . . . . . 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 2805	. . . . . . . . . . 11 |- c e. _V
43	22, 42	op2nda 5221	. . . . . . . . . 10 |- U. ran { <. b , c >. } = c
44	43	eqeq2i 2242	. . . . . . . . 9 |- ( x = U. ran { <. b , c >. } <-> x = c )
45		eqidd 2232	. . . . . . . . . . 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 2292	. . . . . . . . . . 11 |- ( x = c -> ( x e. A <-> c e. A ) )
48		opeq2 3863	. . . . . . . . . . . . 13 |- ( x = c -> <. b , x >. = <. b , c >. )
49	48	sneqd 3682	. . . . . . . . . . . 12 |- ( x = c -> { <. b , x >. } = { <. b , c >. } )
50	49	eqeq2d 2243	. . . . . . . . . . 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 4959	. . . . . . . . . . 11 |- ( a = { <. b , c >. } -> ran a = ran { <. b , c >. } )
55	54	unieqd 3904	. . . . . . . . . 10 |- ( a = { <. b , c >. } -> U. ran a = U. ran { <. b , c >. } )
56	55	eqeq2d 2243	. . . . . . . . 9 |- ( a = { <. b , c >. } -> ( x = U. ran a <-> x = U. ran { <. b , c >. } ) )
57		eqeq1 2238	. . . . . . . . . 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 2644	. . . . . 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 2865	. 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 6225	1 |- ( I e. V -> F : A -1-1-onto-> X_ y e. { I } A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   _Vcvv 2802   {csn 3669   <.cop 3672   U.cuni 3893   |-> cmpt 4150   X. cxp 4723   ran crn 4726   -1-1-onto->wf1o 5325   X_cixp 6866

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ixp 6867

This theorem is referenced by:  mapsnf1o  6905