Theorem fvmptt 5520

Description: Closed theorem form of fvmpt 5506. (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 983	. . 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 5431	. 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 2466	. . . . 5 |- ( A e. D <-> E. x e. D x = A )
4		elex 2700	. . . . . 6 |- ( C e. V -> C e. _V )
5		nfa1 1522	. . . . . . 7 |- F/ x A. x ( x = A -> B = C )
6		nfv 1509	. . . . . . . 8 |- F/ x C e. _V
7		nffvmpt1 5440	. . . . . . . . 9 |- F/_ x ( ( x e. D |-> B ) ` A )
8	7	nfeq1 2292	. . . . . . . 8 |- F/ x ( ( x e. D |-> B ) ` A ) = C
9	6, 8	nfim 1552	. . . . . . 7 |- F/ x ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C )
10		simprl 521	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x e. D )
11		simplr 520	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B = C )
12		simprr 522	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> C e. _V )
13	11, 12	eqeltrd 2217	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B e. _V )
14		eqid 2140	. . . . . . . . . . . . . 14 |- ( x e. D |-> B ) = ( x e. D |-> B )
15	14	fvmpt2 5512	. . . . . . . . . . . . 13 |- ( ( x e. D /\ B e. _V ) -> ( ( x e. D |-> B ) ` x ) = B )
16	10, 13, 15	syl2anc 409	. . . . . . . . . . . 12 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` x ) = B )
17		simpll 519	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x = A )
18	17	fveq2d 5433	. . . . . . . . . . . 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 2181	. . . . . . . . . . 11 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` A ) = C )
20	19	exp43 370	. . . . . . . . . 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 1518	. . . . . . 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 2549	. . . . . 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 1001	. 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 2173	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 963   A.wal 1330   = wceq 1332   e. wcel 1481   E.wrex 2418   _Vcvv 2689   |-> cmpt 3997   ` cfv 5131

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139

This theorem is referenced by: (None)