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Theorem fvmptdv2 5758
Description: Alternate deduction version of fvmpt 5746, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdv2.1  |-  ( ph  ->  A  e.  D )
fvmptdv2.2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
fvmptdv2.3  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
Assertion
Ref Expression
fvmptdv2  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ( 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 fvmptdv2
StepHypRef Expression
1 eqidd 2389 . . 3  |-  ( ph  ->  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B ) )
2 fvmptdv2.3 . . 3  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
3 fvmptdv2.1 . . 3  |-  ( ph  ->  A  e.  D )
4 elex 2908 . . . . . 6  |-  ( A  e.  D  ->  A  e.  _V )
53, 4syl 16 . . . . 5  |-  ( ph  ->  A  e.  _V )
6 isset 2904 . . . . 5  |-  ( A  e.  _V  <->  E. x  x  =  A )
75, 6sylib 189 . . . 4  |-  ( ph  ->  E. x  x  =  A )
8 fvmptdv2.2 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
9 elex 2908 . . . . . 6  |-  ( B  e.  V  ->  B  e.  _V )
108, 9syl 16 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
112, 10eqeltrrd 2463 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  C  e.  _V )
127, 11exlimddv 1645 . . 3  |-  ( ph  ->  C  e.  _V )
131, 2, 3, 12fvmptd 5750 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
14 fveq1 5668 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
1514eqeq1d 2396 . 2  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( ( F `  A )  =  C  <-> 
( ( x  e.  D  |->  B ) `  A )  =  C ) )
1613, 15syl5ibrcom 214 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ( F `  A )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2900    e. cmpt 4208   ` cfv 5395
This theorem is referenced by:  curf12  14252  curf2  14254  yonedalem4b  14301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403
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