Theorem fvmptt 5505

Description: Closed theorem form of fvmpt 5491. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)

Assertion

Ref	Expression
fvmptt	|- ( ( A. x ( x = A -> B = C ) /\ F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> ( F ` A ) = C )

Distinct variable groups:   x, A   x, C   x, D

Allowed substitution hints:   B( x)   F( x)   V( x)

Proof of Theorem fvmptt

Step	Hyp	Ref	Expression
1		simp2 982	. . 3 |- ( ( A. x ( x = A -> B = C ) /\ F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> F = ( x e. D |-> B ) )
2	1	fveq1d 5416	. 2 |- ( ( A. x ( x = A -> B = C ) /\ F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
3		risset 2461	. . . . 5 |- ( A e. D <-> E. x e. D x = A )
4		elex 2692	. . . . . 6 |- ( C e. V -> C e. _V )
5		nfa1 1521	. . . . . . 7 |- F/ x A. x ( x = A -> B = C )
6		nfv 1508	. . . . . . . 8 |- F/ x C e. _V
7		nffvmpt1 5425	. . . . . . . . 9 |- F/_ x ( ( x e. D |-> B ) ` A )
8	7	nfeq1 2289	. . . . . . . 8 |- F/ x ( ( x e. D |-> B ) ` A ) = C
9	6, 8	nfim 1551	. . . . . . 7 |- F/ x ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C )
10		simprl 520	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x e. D )
11		simplr 519	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B = C )
12		simprr 521	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> C e. _V )
13	11, 12	eqeltrd 2214	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B e. _V )
14		eqid 2137	. . . . . . . . . . . . . 14 |- ( x e. D |-> B ) = ( x e. D |-> B )
15	14	fvmpt2 5497	. . . . . . . . . . . . 13 |- ( ( x e. D /\ B e. _V ) -> ( ( x e. D |-> B ) ` x ) = B )
16	10, 13, 15	syl2anc 408	. . . . . . . . . . . 12 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` x ) = B )
17		simpll 518	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x = A )
18	17	fveq2d 5418	. . . . . . . . . . . 12 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` x ) = ( ( x e. D |-> B ) ` A ) )
19	16, 18, 11	3eqtr3d 2178	. . . . . . . . . . 11 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` A ) = C )
20	19	exp43 369	. . . . . . . . . 10 |- ( x = A -> ( B = C -> ( x e. D -> ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C ) ) ) )
21	20	a2i 11	. . . . . . . . 9 |- ( ( x = A -> B = C ) -> ( x = A -> ( x e. D -> ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C ) ) ) )
22	21	com23 78	. . . . . . . 8 |- ( ( x = A -> B = C ) -> ( x e. D -> ( x = A -> ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C ) ) ) )
23	22	sps 1517	. . . . . . 7 |- ( A. x ( x = A -> B = C ) -> ( x e. D -> ( x = A -> ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C ) ) ) )
24	5, 9, 23	rexlimd 2544	. . . . . 6 |- ( A. x ( x = A -> B = C ) -> ( E. x e. D x = A -> ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C ) ) )
25	4, 24	syl7 69	. . . . 5 |- ( A. x ( x = A -> B = C ) -> ( E. x e. D x = A -> ( C e. V -> ( ( x e. D |-> B ) ` A ) = C ) ) )
26	3, 25	syl5bi 151	. . . 4 |- ( A. x ( x = A -> B = C ) -> ( A e. D -> ( C e. V -> ( ( x e. D |-> B ) ` A ) = C ) ) )
27	26	imp32 255	. . 3 |- ( ( A. x ( x = A -> B = C ) /\ ( A e. D /\ C e. V ) ) -> ( ( x e. D |-> B ) ` A ) = C )
28	27	3adant2 1000	. 2 |- ( ( A. x ( x = A -> B = C ) /\ F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> ( ( x e. D |-> B ) ` A ) = C )
29	2, 28	eqtrd 2170	1 |- ( ( A. x ( x = A -> B = C ) /\ F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   A.wal 1329   = wceq 1331   e. wcel 1480   E.wrex 2415   _Vcvv 2681   |-> cmpt 3984   ` cfv 5118

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126

This theorem is referenced by: (None)