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Theorem fvmptd 5470
Description: Deduction version of fvmpt 5466. (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 5391 . 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 3016 . . . 4  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
6 fvmptd.4 . . . 4  |-  ( ph  ->  C  e.  V )
75, 6eqeltrd 2194 . . 3  |-  ( ph  ->  [_ A  /  x ]_ B  e.  V
)
8 eqid 2117 . . . 4  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
98fvmpts 5467 . . 3  |-  ( ( A  e.  D  /\  [_ A  /  x ]_ B  e.  V )  ->  ( ( x  e.  D  |->  B ) `  A )  =  [_ A  /  x ]_ B
)
103, 7, 9syl2anc 408 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  B ) `  A )  =  [_ A  /  x ]_ B
)
112, 10, 53eqtrd 2154 1  |-  ( ph  ->  ( F `  A
)  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   [_csb 2975    |-> cmpt 3959   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101
This theorem is referenced by:  fvmptdv2  5478  rdgivallem  6246  1stinl  6927  2ndinl  6928  1stinr  6929  2ndinr  6930  updjudhcoinlf  6933  updjudhcoinrg  6934  cardcl  7005  caucvgsrlemfv  7567  caucvgsrlemoffval  7572  axcaucvglemval  7673  negiso  8681  infrenegsupex  9357  iseqf1olemfvp  10238  seq3f1olemqsum  10241  infxrnegsupex  11000  climcvg1nlem  11086  isumshft  11227  lmfval  12288  blfvalps  12481  cdivcncfap  12683  peano4nninf  13127  peano3nninf  13128  nninfsellemeq  13137  nninfsellemeqinf  13139
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