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Theorem fvmptdv2 5613
Description: Alternate deduction version of fvmpt 5602, 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 2284 . . 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 2796 . . . . . 6  |-  ( A  e.  D  ->  A  e.  _V )
53, 4syl 15 . . . . 5  |-  ( ph  ->  A  e.  _V )
6 isset 2792 . . . . 5  |-  ( A  e.  _V  <->  E. x  x  =  A )
75, 6sylib 188 . . . 4  |-  ( ph  ->  E. x  x  =  A )
8 fvmptdv2.2 . . . . . . . 8  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
9 elex 2796 . . . . . . . 8  |-  ( B  e.  V  ->  B  e.  _V )
108, 9syl 15 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
112, 10eqeltrrd 2358 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  C  e.  _V )
1211ex 423 . . . . 5  |-  ( ph  ->  ( x  =  A  ->  C  e.  _V ) )
1312exlimdv 1664 . . . 4  |-  ( ph  ->  ( E. x  x  =  A  ->  C  e.  _V ) )
147, 13mpd 14 . . 3  |-  ( ph  ->  C  e.  _V )
151, 2, 3, 14fvmptd 5606 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
16 fveq1 5524 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
1716eqeq1d 2291 . 2  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( ( F `  A )  =  C  <-> 
( ( x  e.  D  |->  B ) `  A )  =  C ) )
1815, 17syl5ibrcom 213 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ( F `  A )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ` cfv 5255
This theorem is referenced by:  curf12  14001  curf2  14003  yonedalem4b  14050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263
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