Theorem fvmptt 5444

Description: Closed theorem form of fvmpt 5430. (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 950	. . 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 5355	. 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 2422	. . . . 5 |- ( A e. D <-> E. x e. D x = A )
4		elex 2652	. . . . . 6 |- ( C e. V -> C e. _V )
5		nfa1 1489	. . . . . . 7 |- F/ x A. x ( x = A -> B = C )
6		nfv 1476	. . . . . . . 8 |- F/ x C e. _V
7		nffvmpt1 5364	. . . . . . . . 9 |- F/_ x ( ( x e. D |-> B ) ` A )
8	7	nfeq1 2250	. . . . . . . 8 |- F/ x ( ( x e. D |-> B ) ` A ) = C
9	6, 8	nfim 1519	. . . . . . 7 |- F/ x ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C )
10		simprl 501	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x e. D )
11		simplr 500	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B = C )
12		simprr 502	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> C e. _V )
13	11, 12	eqeltrd 2176	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B e. _V )
14		eqid 2100	. . . . . . . . . . . . . 14 |- ( x e. D |-> B ) = ( x e. D |-> B )
15	14	fvmpt2 5436	. . . . . . . . . . . . 13 |- ( ( x e. D /\ B e. _V ) -> ( ( x e. D |-> B ) ` x ) = B )
16	10, 13, 15	syl2anc 406	. . . . . . . . . . . 12 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` x ) = B )
17		simpll 499	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x = A )
18	17	fveq2d 5357	. . . . . . . . . . . 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 2140	. . . . . . . . . . 11 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` A ) = C )
20	19	exp43 367	. . . . . . . . . 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 1485	. . . . . . 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 2505	. . . . . 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 968	. 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 2132	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 930   A.wal 1297   = wceq 1299   e. wcel 1448   E.wrex 2376   _Vcvv 2641   |-> cmpt 3929   ` cfv 5059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067

This theorem is referenced by: (None)