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Theorem fvmptdf 5474
Description: Alternate deduction version of fvmpt 5464, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdf.1  |-  ( ph  ->  A  e.  D )
fvmptdf.2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
fvmptdf.3  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  B  ->  ps ) )
fvmptdf.4  |-  F/_ x F
fvmptdf.5  |-  F/ x ps
Assertion
Ref Expression
fvmptdf  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Distinct variable groups:    x, A    x, D    ph, x
Allowed substitution hints:    ps( x)    B( x)    F( x)    V( x)

Proof of Theorem fvmptdf
StepHypRef Expression
1 nfv 1491 . 2  |-  F/ x ph
2 fvmptdf.4 . . . 4  |-  F/_ x F
3 nfmpt1 3989 . . . 4  |-  F/_ x
( x  e.  D  |->  B )
42, 3nfeq 2264 . . 3  |-  F/ x  F  =  ( x  e.  D  |->  B )
5 fvmptdf.5 . . 3  |-  F/ x ps
64, 5nfim 1534 . 2  |-  F/ x
( F  =  ( x  e.  D  |->  B )  ->  ps )
7 fvmptdf.1 . . . 4  |-  ( ph  ->  A  e.  D )
8 elex 2669 . . . 4  |-  ( A  e.  D  ->  A  e.  _V )
97, 8syl 14 . . 3  |-  ( ph  ->  A  e.  _V )
10 isset 2664 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
119, 10sylib 121 . 2  |-  ( ph  ->  E. x  x  =  A )
12 fveq1 5386 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
13 simpr 109 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
1413fveq2d 5391 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  ( ( x  e.  D  |->  B ) `  A ) )
157adantr 272 . . . . . . . 8  |-  ( (
ph  /\  x  =  A )  ->  A  e.  D )
1613, 15eqeltrd 2192 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  e.  D )
17 fvmptdf.2 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
18 eqid 2115 . . . . . . . 8  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1918fvmpt2 5470 . . . . . . 7  |-  ( ( x  e.  D  /\  B  e.  V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
2016, 17, 19syl2anc 406 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  B )
2114, 20eqtr3d 2150 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  A
)  =  B )
2221eqeq2d 2127 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A )  <-> 
( F `  A
)  =  B ) )
23 fvmptdf.3 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  B  ->  ps ) )
2422, 23sylbid 149 . . 3  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A )  ->  ps ) )
2512, 24syl5 32 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) )
261, 6, 11, 25exlimdd 1826 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314   F/wnf 1419   E.wex 1451    e. wcel 1463   F/_wnfc 2243   _Vcvv 2658    |-> cmpt 3957   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099
This theorem is referenced by:  fvmptdv  5475
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