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Theorem fvmptdf 5961
Description: Alternate deduction version of fvmpt 5948, 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 1761 . 2  |-  F/ x ph
2 fvmptdf.4 . . . 4  |-  F/_ x F
3 nfmpt1 4492 . . . 4  |-  F/_ x
( x  e.  D  |->  B )
42, 3nfeq 2603 . . 3  |-  F/ x  F  =  ( x  e.  D  |->  B )
5 fvmptdf.5 . . 3  |-  F/ x ps
64, 5nfim 2003 . 2  |-  F/ x
( F  =  ( x  e.  D  |->  B )  ->  ps )
7 fvmptdf.1 . . . 4  |-  ( ph  ->  A  e.  D )
8 elex 3054 . . . 4  |-  ( A  e.  D  ->  A  e.  _V )
97, 8syl 17 . . 3  |-  ( ph  ->  A  e.  _V )
10 isset 3049 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
119, 10sylib 200 . 2  |-  ( ph  ->  E. x  x  =  A )
12 fveq1 5864 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
13 simpr 463 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
1413fveq2d 5869 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  ( ( x  e.  D  |->  B ) `  A ) )
157adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  =  A )  ->  A  e.  D )
1613, 15eqeltrd 2529 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  e.  D )
17 fvmptdf.2 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
18 eqid 2451 . . . . . . . 8  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1918fvmpt2 5957 . . . . . . 7  |-  ( ( x  e.  D  /\  B  e.  V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
2016, 17, 19syl2anc 667 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  B )
2114, 20eqtr3d 2487 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  A
)  =  B )
2221eqeq2d 2461 . . . 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 219 . . 3  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A )  ->  ps ) )
2512, 24syl5 33 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) )
261, 6, 11, 25exlimdd 2070 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444   E.wex 1663   F/wnf 1667    e. wcel 1887   F/_wnfc 2579   _Vcvv 3045    |-> cmpt 4461   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590
This theorem is referenced by:  fvmptdv  5962  yonedalem4b  16161
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