Theorem fvmptd 5454

Description: Deduction version of fvmpt 5450. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
fvmptd.1	|- ( ph -> F = ( x e. D |-> B ) )
fvmptd.2	|- ( ( ph /\ x = A ) -> B = C )
fvmptd.3	|- ( ph -> A e. D )
fvmptd.4	|- ( ph -> C e. V )

Assertion

Ref	Expression
fvmptd	|- ( ph -> ( F ` A ) = C )

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

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

Proof of Theorem fvmptd

Step	Hyp	Ref	Expression
1		fvmptd.1	. . 3 |- ( ph -> F = ( x e. D |-> B ) )
2	1	fveq1d 5375	. 2 |- ( ph -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
3		fvmptd.3	. . 3 |- ( ph -> A e. D )
4		fvmptd.2	. . . . 5 |- ( ( ph /\ x = A ) -> B = C )
5	3, 4	csbied 3010	. . . 4 |- ( ph -> [_ A / x ]_ B = C )
6		fvmptd.4	. . . 4 |- ( ph -> C e. V )
7	5, 6	eqeltrd 2189	. . 3 |- ( ph -> [_ A / x ]_ B e. V )
8		eqid 2113	. . . 4 |- ( x e. D |-> B ) = ( x e. D |-> B )
9	8	fvmpts 5451	. . 3 |- ( ( A e. D /\ [_ A / x ]_ B e. V ) -> ( ( x e. D |-> B ) ` A ) = [_ A / x ]_ B )
10	3, 7, 9	syl2anc 406	. 2 |- ( ph -> ( ( x e. D |-> B ) ` A ) = [_ A / x ]_ B )
11	2, 10, 5	3eqtrd 2149	1 |- ( ph -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1312   e. wcel 1461   [_csb 2969   |-> cmpt 3947   ` cfv 5079

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 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087

This theorem is referenced by:  fvmptdv2  5462  rdgivallem  6230  1stinl  6909  2ndinl  6910  1stinr  6911  2ndinr  6912  updjudhcoinlf  6915  updjudhcoinrg  6916  cardcl  6984  caucvgsrlemfv  7527  caucvgsrlemoffval  7532  axcaucvglemval  7626  negiso  8617  infrenegsupex  9285  iseqf1olemfvp  10157  seq3f1olemqsum  10160  infxrnegsupex  10918  climcvg1nlem  11004  isumshft  11145  lmfval  12198  blfvalps  12368  cdivcncfap  12567  peano4nninf  12881  peano3nninf  12882  nninfsellemeq  12891  nninfsellemeqinf  12893