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Theorem fvmptdv2 5629
Description: Alternate deduction version of fvmpt 5618, 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 2297 . . 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 2809 . . . . . 6  |-  ( A  e.  D  ->  A  e.  _V )
53, 4syl 15 . . . . 5  |-  ( ph  ->  A  e.  _V )
6 isset 2805 . . . . 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 2809 . . . . . . . 8  |-  ( B  e.  V  ->  B  e.  _V )
108, 9syl 15 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
112, 10eqeltrrd 2371 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  C  e.  _V )
1211ex 423 . . . . 5  |-  ( ph  ->  ( x  =  A  ->  C  e.  _V ) )
1312exlimdv 1626 . . . 4  |-  ( ph  ->  ( E. x  x  =  A  ->  C  e.  _V ) )
147, 13mpd 14 . . 3  |-  ( ph  ->  C  e.  _V )
151, 2, 3, 14fvmptd 5622 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
16 fveq1 5540 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
1716eqeq1d 2304 . 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 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   ` cfv 5271
This theorem is referenced by:  curf12  14017  curf2  14019  yonedalem4b  14066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279
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