Theorem fliftfun 5456

Description: The function F is the unique function defined by F ` A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	|- F = ran ( x e. X |-> <. A , B >. )
flift.2	|- ( ( ph /\ x e. X ) -> A e. R )
flift.3	|- ( ( ph /\ x e. X ) -> B e. S )
fliftfun.4	|- ( x = y -> A = C )
fliftfun.5	|- ( x = y -> B = D )

Assertion

Ref	Expression
fliftfun	|- ( ph -> ( Fun F <-> A. x e. X A. y e. X ( A = C -> B = D ) ) )

Distinct variable groups:   y, A   y, B   x, C   x, y, R   x, D   y, F   ph, x, y   x, X, y   x, S, y

Allowed substitution hints:   A( x)   B( x)   C( y)   D( y)   F( x)

Proof of Theorem fliftfun

Dummy variables v u z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . 3 |- F/ x ph
2		flift.1	. . . . 5 |- F = ran ( x e. X |-> <. A , B >. )
3		nfmpt1 3871	. . . . . 6 |- F/_ x ( x e. X |-> <. A , B >. )
4	3	nfrn 4597	. . . . 5 |- F/_ x ran ( x e. X |-> <. A , B >. )
5	2, 4	nfcxfr 2216	. . . 4 |- F/_ x F
6	5	nffun 4944	. . 3 |- F/ x Fun F
7		fveq2 5198	. . . . . . 7 |- ( A = C -> ( F ` A ) = ( F ` C ) )
8		simplr 496	. . . . . . . . 9 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> Fun F )
9		flift.2	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> A e. R )
10		flift.3	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> B e. S )
11	2, 9, 10	fliftel1 5454	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> A F B )
12	11	ad2ant2r 492	. . . . . . . . 9 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> A F B )
13		funbrfv 5233	. . . . . . . . 9 |- ( Fun F -> ( A F B -> ( F ` A ) = B ) )
14	8, 12, 13	sylc 61	. . . . . . . 8 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> ( F ` A ) = B )
15		simprr 498	. . . . . . . . . . 11 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> y e. X )
16		eqidd 2082	. . . . . . . . . . 11 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> C = C )
17		eqidd 2082	. . . . . . . . . . 11 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> D = D )
18		fliftfun.4	. . . . . . . . . . . . . 14 |- ( x = y -> A = C )
19	18	eqeq2d 2092	. . . . . . . . . . . . 13 |- ( x = y -> ( C = A <-> C = C ) )
20		fliftfun.5	. . . . . . . . . . . . . 14 |- ( x = y -> B = D )
21	20	eqeq2d 2092	. . . . . . . . . . . . 13 |- ( x = y -> ( D = B <-> D = D ) )
22	19, 21	anbi12d 456	. . . . . . . . . . . 12 |- ( x = y -> ( ( C = A /\ D = B ) <-> ( C = C /\ D = D ) ) )
23	22	rspcev 2701	. . . . . . . . . . 11 |- ( ( y e. X /\ ( C = C /\ D = D ) ) -> E. x e. X ( C = A /\ D = B ) )
24	15, 16, 17, 23	syl12anc 1167	. . . . . . . . . 10 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> E. x e. X ( C = A /\ D = B ) )
25	2, 9, 10	fliftel 5453	. . . . . . . . . . 11 |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )
26	25	ad2antrr 471	. . . . . . . . . 10 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )
27	24, 26	mpbird 165	. . . . . . . . 9 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> C F D )
28		funbrfv 5233	. . . . . . . . 9 |- ( Fun F -> ( C F D -> ( F ` C ) = D ) )
29	8, 27, 28	sylc 61	. . . . . . . 8 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> ( F ` C ) = D )
30	14, 29	eqeq12d 2095	. . . . . . 7 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> ( ( F ` A ) = ( F ` C ) <-> B = D ) )
31	7, 30	syl5ib 152	. . . . . 6 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> ( A = C -> B = D ) )
32	31	anassrs 392	. . . . 5 |- ( ( ( ( ph /\ Fun F ) /\ x e. X ) /\ y e. X ) -> ( A = C -> B = D ) )
33	32	ralrimiva 2434	. . . 4 |- ( ( ( ph /\ Fun F ) /\ x e. X ) -> A. y e. X ( A = C -> B = D ) )
34	33	exp31 356	. . 3 |- ( ph -> ( Fun F -> ( x e. X -> A. y e. X ( A = C -> B = D ) ) ) )
35	1, 6, 34	ralrimd 2439	. 2 |- ( ph -> ( Fun F -> A. x e. X A. y e. X ( A = C -> B = D ) ) )
36	2, 9, 10	fliftel 5453	. . . . . . . . 9 |- ( ph -> ( z F u <-> E. x e. X ( z = A /\ u = B ) ) )
37	2, 9, 10	fliftel 5453	. . . . . . . . . 10 |- ( ph -> ( z F v <-> E. x e. X ( z = A /\ v = B ) ) )
38	18	eqeq2d 2092	. . . . . . . . . . . 12 |- ( x = y -> ( z = A <-> z = C ) )
39	20	eqeq2d 2092	. . . . . . . . . . . 12 |- ( x = y -> ( v = B <-> v = D ) )
40	38, 39	anbi12d 456	. . . . . . . . . . 11 |- ( x = y -> ( ( z = A /\ v = B ) <-> ( z = C /\ v = D ) ) )
41	40	cbvrexv 2578	. . . . . . . . . 10 |- ( E. x e. X ( z = A /\ v = B ) <-> E. y e. X ( z = C /\ v = D ) )
42	37, 41	syl6bb 194	. . . . . . . . 9 |- ( ph -> ( z F v <-> E. y e. X ( z = C /\ v = D ) ) )
43	36, 42	anbi12d 456	. . . . . . . 8 |- ( ph -> ( ( z F u /\ z F v ) <-> ( E. x e. X ( z = A /\ u = B ) /\ E. y e. X ( z = C /\ v = D ) ) ) )
44	43	biimpd 142	. . . . . . 7 |- ( ph -> ( ( z F u /\ z F v ) -> ( E. x e. X ( z = A /\ u = B ) /\ E. y e. X ( z = C /\ v = D ) ) ) )
45		reeanv 2523	. . . . . . . 8 |- ( E. x e. X E. y e. X ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) <-> ( E. x e. X ( z = A /\ u = B ) /\ E. y e. X ( z = C /\ v = D ) ) )
46		r19.29 2494	. . . . . . . . . 10 |- ( ( A. x e. X A. y e. X ( A = C -> B = D ) /\ E. x e. X E. y e. X ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> E. x e. X ( A. y e. X ( A = C -> B = D ) /\ E. y e. X ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) )
47		r19.29 2494	. . . . . . . . . . . 12 |- ( ( A. y e. X ( A = C -> B = D ) /\ E. y e. X ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> E. y e. X ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) )
48		eqtr2 2099	. . . . . . . . . . . . . . . . 17 |- ( ( z = A /\ z = C ) -> A = C )
49	48	ad2ant2r 492	. . . . . . . . . . . . . . . 16 |- ( ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) -> A = C )
50	49	imim1i 59	. . . . . . . . . . . . . . 15 |- ( ( A = C -> B = D ) -> ( ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) -> B = D ) )
51	50	imp 122	. . . . . . . . . . . . . 14 |- ( ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> B = D )
52		simprlr 504	. . . . . . . . . . . . . 14 |- ( ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> u = B )
53		simprrr 506	. . . . . . . . . . . . . 14 |- ( ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> v = D )
54	51, 52, 53	3eqtr4d 2123	. . . . . . . . . . . . 13 |- ( ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> u = v )
55	54	rexlimivw 2473	. . . . . . . . . . . 12 |- ( E. y e. X ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> u = v )
56	47, 55	syl 14	. . . . . . . . . . 11 |- ( ( A. y e. X ( A = C -> B = D ) /\ E. y e. X ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> u = v )
57	56	rexlimivw 2473	. . . . . . . . . 10 |- ( E. x e. X ( A. y e. X ( A = C -> B = D ) /\ E. y e. X ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> u = v )
58	46, 57	syl 14	. . . . . . . . 9 |- ( ( A. x e. X A. y e. X ( A = C -> B = D ) /\ E. x e. X E. y e. X ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> u = v )
59	58	ex 113	. . . . . . . 8 |- ( A. x e. X A. y e. X ( A = C -> B = D ) -> ( E. x e. X E. y e. X ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) -> u = v ) )
60	45, 59	syl5bir 151	. . . . . . 7 |- ( A. x e. X A. y e. X ( A = C -> B = D ) -> ( ( E. x e. X ( z = A /\ u = B ) /\ E. y e. X ( z = C /\ v = D ) ) -> u = v ) )
61	44, 60	syl9 71	. . . . . 6 |- ( ph -> ( A. x e. X A. y e. X ( A = C -> B = D ) -> ( ( z F u /\ z F v ) -> u = v ) ) )
62	61	alrimdv 1797	. . . . 5 |- ( ph -> ( A. x e. X A. y e. X ( A = C -> B = D ) -> A. v ( ( z F u /\ z F v ) -> u = v ) ) )
63	62	alrimdv 1797	. . . 4 |- ( ph -> ( A. x e. X A. y e. X ( A = C -> B = D ) -> A. u A. v ( ( z F u /\ z F v ) -> u = v ) ) )
64	63	alrimdv 1797	. . 3 |- ( ph -> ( A. x e. X A. y e. X ( A = C -> B = D ) -> A. z A. u A. v ( ( z F u /\ z F v ) -> u = v ) ) )
65	2, 9, 10	fliftrel 5452	. . . . 5 |- ( ph -> F C_ ( R X. S ) )
66		relxp 4465	. . . . 5 |- Rel ( R X. S )
67		relss 4445	. . . . 5 |- ( F C_ ( R X. S ) -> ( Rel ( R X. S ) -> Rel F ) )
68	65, 66, 67	mpisyl 1375	. . . 4 |- ( ph -> Rel F )
69		dffun2 4932	. . . . 5 |- ( Fun F <-> ( Rel F /\ A. z A. u A. v ( ( z F u /\ z F v ) -> u = v ) ) )
70	69	baib 861	. . . 4 |- ( Rel F -> ( Fun F <-> A. z A. u A. v ( ( z F u /\ z F v ) -> u = v ) ) )
71	68, 70	syl 14	. . 3 |- ( ph -> ( Fun F <-> A. z A. u A. v ( ( z F u /\ z F v ) -> u = v ) ) )
72	64, 71	sylibrd 167	. 2 |- ( ph -> ( A. x e. X A. y e. X ( A = C -> B = D ) -> Fun F ) )
73	35, 72	impbid 127	1 |- ( ph -> ( Fun F <-> A. x e. X A. y e. X ( A = C -> B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   A.wral 2348   E.wrex 2349   C_ wss 2973   <.cop 3401   class class class wbr 3785   |-> cmpt 3839   X. cxp 4361   ran crn 4364   Rel wrel 4368   Fun wfun 4916   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by:  fliftfund  5457  fliftfuns  5458  qliftfun  6211