Theorem fliftfun 5775

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 1521	. . 3 |- F/ x ph
2		flift.1	. . . . 5 |- F = ran ( x e. X |-> <. A , B >. )
3		nfmpt1 4082	. . . . . 6 |- F/_ x ( x e. X |-> <. A , B >. )
4	3	nfrn 4856	. . . . 5 |- F/_ x ran ( x e. X |-> <. A , B >. )
5	2, 4	nfcxfr 2309	. . . 4 |- F/_ x F
6	5	nffun 5221	. . 3 |- F/ x Fun F
7		fveq2 5496	. . . . . . 7 |- ( A = C -> ( F ` A ) = ( F ` C ) )
8		simplr 525	. . . . . . . . 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 5773	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> A F B )
12	11	ad2ant2r 506	. . . . . . . . 9 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> A F B )
13		funbrfv 5535	. . . . . . . . 9 |- ( Fun F -> ( A F B -> ( F ` A ) = B ) )
14	8, 12, 13	sylc 62	. . . . . . . 8 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> ( F ` A ) = B )
15		simprr 527	. . . . . . . . . . 11 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> y e. X )
16		eqidd 2171	. . . . . . . . . . 11 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> C = C )
17		eqidd 2171	. . . . . . . . . . 11 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> D = D )
18		fliftfun.4	. . . . . . . . . . . . . 14 |- ( x = y -> A = C )
19	18	eqeq2d 2182	. . . . . . . . . . . . 13 |- ( x = y -> ( C = A <-> C = C ) )
20		fliftfun.5	. . . . . . . . . . . . . 14 |- ( x = y -> B = D )
21	20	eqeq2d 2182	. . . . . . . . . . . . 13 |- ( x = y -> ( D = B <-> D = D ) )
22	19, 21	anbi12d 470	. . . . . . . . . . . 12 |- ( x = y -> ( ( C = A /\ D = B ) <-> ( C = C /\ D = D ) ) )
23	22	rspcev 2834	. . . . . . . . . . 11 |- ( ( y e. X /\ ( C = C /\ D = D ) ) -> E. x e. X ( C = A /\ D = B ) )
24	15, 16, 17, 23	syl12anc 1231	. . . . . . . . . 10 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> E. x e. X ( C = A /\ D = B ) )
25	2, 9, 10	fliftel 5772	. . . . . . . . . . 11 |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )
26	25	ad2antrr 485	. . . . . . . . . 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 166	. . . . . . . . 9 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> C F D )
28		funbrfv 5535	. . . . . . . . 9 |- ( Fun F -> ( C F D -> ( F ` C ) = D ) )
29	8, 27, 28	sylc 62	. . . . . . . 8 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> ( F ` C ) = D )
30	14, 29	eqeq12d 2185	. . . . . . 7 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> ( ( F ` A ) = ( F ` C ) <-> B = D ) )
31	7, 30	syl5ib 153	. . . . . 6 |- ( ( ( ph /\ Fun F ) /\ ( x e. X /\ y e. X ) ) -> ( A = C -> B = D ) )
32	31	anassrs 398	. . . . 5 |- ( ( ( ( ph /\ Fun F ) /\ x e. X ) /\ y e. X ) -> ( A = C -> B = D ) )
33	32	ralrimiva 2543	. . . 4 |- ( ( ( ph /\ Fun F ) /\ x e. X ) -> A. y e. X ( A = C -> B = D ) )
34	33	exp31 362	. . 3 |- ( ph -> ( Fun F -> ( x e. X -> A. y e. X ( A = C -> B = D ) ) ) )
35	1, 6, 34	ralrimd 2548	. 2 |- ( ph -> ( Fun F -> A. x e. X A. y e. X ( A = C -> B = D ) ) )
36	2, 9, 10	fliftel 5772	. . . . . . . . 9 |- ( ph -> ( z F u <-> E. x e. X ( z = A /\ u = B ) ) )
37	2, 9, 10	fliftel 5772	. . . . . . . . . 10 |- ( ph -> ( z F v <-> E. x e. X ( z = A /\ v = B ) ) )
38	18	eqeq2d 2182	. . . . . . . . . . . 12 |- ( x = y -> ( z = A <-> z = C ) )
39	20	eqeq2d 2182	. . . . . . . . . . . 12 |- ( x = y -> ( v = B <-> v = D ) )
40	38, 39	anbi12d 470	. . . . . . . . . . 11 |- ( x = y -> ( ( z = A /\ v = B ) <-> ( z = C /\ v = D ) ) )
41	40	cbvrexv 2697	. . . . . . . . . 10 |- ( E. x e. X ( z = A /\ v = B ) <-> E. y e. X ( z = C /\ v = D ) )
42	37, 41	bitrdi 195	. . . . . . . . 9 |- ( ph -> ( z F v <-> E. y e. X ( z = C /\ v = D ) ) )
43	36, 42	anbi12d 470	. . . . . . . 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 143	. . . . . . 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 2639	. . . . . . . 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 2607	. . . . . . . . . 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 2607	. . . . . . . . . . . 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 2189	. . . . . . . . . . . . . . . . 17 |- ( ( z = A /\ z = C ) -> A = C )
49	48	ad2ant2r 506	. . . . . . . . . . . . . . . 16 |- ( ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) -> A = C )
50	49	imim1i 60	. . . . . . . . . . . . . . 15 |- ( ( A = C -> B = D ) -> ( ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) -> B = D ) )
51	50	imp 123	. . . . . . . . . . . . . 14 |- ( ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> B = D )
52		simprlr 533	. . . . . . . . . . . . . 14 |- ( ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> u = B )
53		simprrr 535	. . . . . . . . . . . . . 14 |- ( ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> v = D )
54	51, 52, 53	3eqtr4d 2213	. . . . . . . . . . . . 13 |- ( ( ( A = C -> B = D ) /\ ( ( z = A /\ u = B ) /\ ( z = C /\ v = D ) ) ) -> u = v )
55	54	rexlimivw 2583	. . . . . . . . . . . 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 2583	. . . . . . . . . 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 114	. . . . . . . 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 152	. . . . . . 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 72	. . . . . 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 1869	. . . . 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 1869	. . . 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 1869	. . 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 5771	. . . . 5 |- ( ph -> F C_ ( R X. S ) )
66		relxp 4720	. . . . 5 |- Rel ( R X. S )
67		relss 4698	. . . . 5 |- ( F C_ ( R X. S ) -> ( Rel ( R X. S ) -> Rel F ) )
68	65, 66, 67	mpisyl 1439	. . . 4 |- ( ph -> Rel F )
69		dffun2 5208	. . . . 5 |- ( Fun F <-> ( Rel F /\ A. z A. u A. v ( ( z F u /\ z F v ) -> u = v ) ) )
70	69	baib 914	. . . 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 168	. 2 |- ( ph -> ( A. x e. X A. y e. X ( A = C -> B = D ) -> Fun F ) )
73	35, 72	impbid 128	1 |- ( ph -> ( Fun F <-> A. x e. X A. y e. X ( A = C -> B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   C_ wss 3121   <.cop 3586   class class class wbr 3989   |-> cmpt 4050   X. cxp 4609   ran crn 4612   Rel wrel 4616   Fun wfun 5192   ` cfv 5198

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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

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-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  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-fv 5206

This theorem is referenced by:  fliftfund  5776  fliftfuns  5777  qliftfun  6595