Theorem fvmptt 6300

Description: Closed theorem form of fvmpt 6282. (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 1062	. . 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 6193	. 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 3062	. . . . 5 |- ( A e. D <-> E. x e. D x = A )
4		elex 3212	. . . . . 6 |- ( C e. V -> C e. _V )
5		nfa1 2028	. . . . . . 7 |- F/ x A. x ( x = A -> B = C )
6		nfv 1843	. . . . . . . 8 |- F/ x C e. _V
7		nffvmpt1 6199	. . . . . . . . 9 |- F/_ x ( ( x e. D |-> B ) ` A )
8	7	nfeq1 2778	. . . . . . . 8 |- F/ x ( ( x e. D |-> B ) ` A ) = C
9	6, 8	nfim 1825	. . . . . . 7 |- F/ x ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C )
10		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x e. D )
11		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B = C )
12		simprr 796	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> C e. _V )
13	11, 12	eqeltrd 2701	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B e. _V )
14		eqid 2622	. . . . . . . . . . . . . 14 |- ( x e. D |-> B ) = ( x e. D |-> B )
15	14	fvmpt2 6291	. . . . . . . . . . . . 13 |- ( ( x e. D /\ B e. _V ) -> ( ( x e. D |-> B ) ` x ) = B )
16	10, 13, 15	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` x ) = B )
17		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x = A )
18	17	fveq2d 6195	. . . . . . . . . . . 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 2664	. . . . . . . . . . 11 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` A ) = C )
20	19	exp43 640	. . . . . . . . . 10 |- ( x = A -> ( B = C -> ( x e. D -> ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C ) ) ) )
21	20	a2i 14	. . . . . . . . 9 |- ( ( x = A -> B = C ) -> ( x = A -> ( x e. D -> ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C ) ) ) )
22	21	com23 86	. . . . . . . 8 |- ( ( x = A -> B = C ) -> ( x e. D -> ( x = A -> ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C ) ) ) )
23	22	sps 2055	. . . . . . 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 3026	. . . . . 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 74	. . . . 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 232	. . . 4 |- ( A. x ( x = A -> B = C ) -> ( A e. D -> ( C e. V -> ( ( x e. D |-> B ) ` A ) = C ) ) )
27	26	imp32 449	. . 3 |- ( ( A. x ( x = A -> B = C ) /\ ( A e. D /\ C e. V ) ) -> ( ( x e. D |-> B ) ` A ) = C )
28	27	3adant2 1080	. 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 2656	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 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by: (None)