Theorem fvmptt 5649

Description: Closed theorem form of fvmpt 5634. (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 1000	. . 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 5556	. 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 2522	. . . . 5 |- ( A e. D <-> E. x e. D x = A )
4		elex 2771	. . . . . 6 |- ( C e. V -> C e. _V )
5		nfa1 1552	. . . . . . 7 |- F/ x A. x ( x = A -> B = C )
6		nfv 1539	. . . . . . . 8 |- F/ x C e. _V
7		nffvmpt1 5565	. . . . . . . . 9 |- F/_ x ( ( x e. D |-> B ) ` A )
8	7	nfeq1 2346	. . . . . . . 8 |- F/ x ( ( x e. D |-> B ) ` A ) = C
9	6, 8	nfim 1583	. . . . . . 7 |- F/ x ( C e. _V -> ( ( x e. D |-> B ) ` A ) = C )
10		simprl 529	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x e. D )
11		simplr 528	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B = C )
12		simprr 531	. . . . . . . . . . . . . 14 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> C e. _V )
13	11, 12	eqeltrd 2270	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> B e. _V )
14		eqid 2193	. . . . . . . . . . . . . 14 |- ( x e. D |-> B ) = ( x e. D |-> B )
15	14	fvmpt2 5641	. . . . . . . . . . . . 13 |- ( ( x e. D /\ B e. _V ) -> ( ( x e. D |-> B ) ` x ) = B )
16	10, 13, 15	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` x ) = B )
17		simpll 527	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> x = A )
18	17	fveq2d 5558	. . . . . . . . . . . 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 2234	. . . . . . . . . . 11 |- ( ( ( x = A /\ B = C ) /\ ( x e. D /\ C e. _V ) ) -> ( ( x e. D |-> B ) ` A ) = C )
20	19	exp43 372	. . . . . . . . . 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 1548	. . . . . . 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 2608	. . . . . 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	biimtrid 152	. . . 4 |- ( A. x ( x = A -> B = C ) -> ( A e. D -> ( C e. V -> ( ( x e. D |-> B ) ` A ) = C ) ) )
27	26	imp32 257	. . 3 |- ( ( A. x ( x = A -> B = C ) /\ ( A e. D /\ C e. V ) ) -> ( ( x e. D |-> B ) ` A ) = C )
28	27	3adant2 1018	. 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 2226	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 104   /\ w3a 980   A.wal 1362   = wceq 1364   e. wcel 2164   E.wrex 2473   _Vcvv 2760   |-> cmpt 4090   ` cfv 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262

This theorem is referenced by: (None)