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Theorem fvmptdv2 5575
Description: Alternate deduction version of fvmpt 5564, 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 2285 . . 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 2797 . . . . . 6  |-  ( A  e.  D  ->  A  e.  _V )
53, 4syl 15 . . . . 5  |-  ( ph  ->  A  e.  _V )
6 isset 2793 . . . . 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 2797 . . . . . . . 8  |-  ( B  e.  V  ->  B  e.  _V )
108, 9syl 15 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
112, 10eqeltrrd 2359 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  C  e.  _V )
1211ex 423 . . . . 5  |-  ( ph  ->  ( x  =  A  ->  C  e.  _V ) )
1312exlimdv 1665 . . . 4  |-  ( ph  ->  ( E. x  x  =  A  ->  C  e.  _V ) )
147, 13mpd 14 . . 3  |-  ( ph  ->  C  e.  _V )
151, 2, 3, 14fvmptd 5568 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
16 fveq1 5485 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
1716eqeq1d 2292 . 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 1685   _Vcvv 2789    e. cmpt 4078   ` cfv 5221
This theorem is referenced by:  curf12  13997  curf2  13999  yonedalem4b  14046
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 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229
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