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Theorem fvmptd 5285
Description: Deduction version of fvmpt 5281. (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
StepHypRef Expression
1 fvmptd.1 . . 3  |-  ( ph  ->  F  =  ( x  e.  D  |->  B ) )
21fveq1d 5211 . 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 )
53, 4csbied 2949 . . . 4  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
6 fvmptd.4 . . . 4  |-  ( ph  ->  C  e.  V )
75, 6eqeltrd 2156 . . 3  |-  ( ph  ->  [_ A  /  x ]_ B  e.  V
)
8 eqid 2082 . . . 4  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
98fvmpts 5282 . . 3  |-  ( ( A  e.  D  /\  [_ A  /  x ]_ B  e.  V )  ->  ( ( x  e.  D  |->  B ) `  A )  =  [_ A  /  x ]_ B
)
103, 7, 9syl2anc 403 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  B ) `  A )  =  [_ A  /  x ]_ B
)
112, 10, 53eqtrd 2118 1  |-  ( ph  ->  ( F `  A
)  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   [_csb 2909    |-> cmpt 3847   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940
This theorem is referenced by:  fvmptdv2  5292  rdgivallem  6030  cardcl  6509  caucvgsrlemfv  7029  caucvgsrlemoffval  7034  axcaucvglemval  7125  negiso  8100  infrenegsupex  8763  climcvg1nlem  10324
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