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Theorem fvmptdv2 5785
Description: Alternate deduction version of fvmpt 5772, 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 2442 . . 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 2979 . . . . . 6  |-  ( A  e.  D  ->  A  e.  _V )
53, 4syl 16 . . . . 5  |-  ( ph  ->  A  e.  _V )
6 isset 2974 . . . . 5  |-  ( A  e.  _V  <->  E. x  x  =  A )
75, 6sylib 196 . . . 4  |-  ( ph  ->  E. x  x  =  A )
8 fvmptdv2.2 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
9 elex 2979 . . . . . 6  |-  ( B  e.  V  ->  B  e.  _V )
108, 9syl 16 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
112, 10eqeltrrd 2516 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  C  e.  _V )
127, 11exlimddv 1692 . . 3  |-  ( ph  ->  C  e.  _V )
131, 2, 3, 12fvmptd 5777 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
14 fveq1 5688 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
1514eqeq1d 2449 . 2  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( ( F `  A )  =  C  <-> 
( ( x  e.  D  |->  B ) `  A )  =  C ) )
1613, 15syl5ibrcom 222 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ( F `  A )  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2970    e. cmpt 4348   ` cfv 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424
This theorem is referenced by:  curf12  15035  curf2  15037  yonedalem4b  15084
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