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Theorem fvmptdf 5624
Description: Alternate deduction version of fvmpt 5614, 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 1539 . 2  |-  F/ x ph
2 fvmptdf.4 . . . 4  |-  F/_ x F
3 nfmpt1 4111 . . . 4  |-  F/_ x
( x  e.  D  |->  B )
42, 3nfeq 2340 . . 3  |-  F/ x  F  =  ( x  e.  D  |->  B )
5 fvmptdf.5 . . 3  |-  F/ x ps
64, 5nfim 1583 . 2  |-  F/ x
( F  =  ( x  e.  D  |->  B )  ->  ps )
7 fvmptdf.1 . . . 4  |-  ( ph  ->  A  e.  D )
8 elex 2763 . . . 4  |-  ( A  e.  D  ->  A  e.  _V )
97, 8syl 14 . . 3  |-  ( ph  ->  A  e.  _V )
10 isset 2758 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
119, 10sylib 122 . 2  |-  ( ph  ->  E. x  x  =  A )
12 fveq1 5533 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
13 simpr 110 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
1413fveq2d 5538 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  ( ( x  e.  D  |->  B ) `  A ) )
157adantr 276 . . . . . . . 8  |-  ( (
ph  /\  x  =  A )  ->  A  e.  D )
1613, 15eqeltrd 2266 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  e.  D )
17 fvmptdf.2 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
18 eqid 2189 . . . . . . . 8  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1918fvmpt2 5620 . . . . . . 7  |-  ( ( x  e.  D  /\  B  e.  V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
2016, 17, 19syl2anc 411 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  B )
2114, 20eqtr3d 2224 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  A
)  =  B )
2221eqeq2d 2201 . . . 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 150 . . 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 1883 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364   F/wnf 1471   E.wex 1503    e. wcel 2160   F/_wnfc 2319   _Vcvv 2752    |-> cmpt 4079   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243
This theorem is referenced by:  fvmptdv  5625
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