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Theorem fvmptdf 5611
Description: Alternate deduction version of fvmpt 5602, 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 fvmptdf.1 . . . 4  |-  ( ph  ->  A  e.  D )
2 elex 2796 . . . 4  |-  ( A  e.  D  ->  A  e.  _V )
31, 2syl 15 . . 3  |-  ( ph  ->  A  e.  _V )
4 isset 2792 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
53, 4sylib 188 . 2  |-  ( ph  ->  E. x  x  =  A )
6 nfv 1605 . . 3  |-  F/ x ph
7 fvmptdf.4 . . . . 5  |-  F/_ x F
8 nfmpt1 4109 . . . . 5  |-  F/_ x
( x  e.  D  |->  B )
97, 8nfeq 2426 . . . 4  |-  F/ x  F  =  ( x  e.  D  |->  B )
10 fvmptdf.5 . . . 4  |-  F/ x ps
119, 10nfim 1769 . . 3  |-  F/ x
( F  =  ( x  e.  D  |->  B )  ->  ps )
12 fveq1 5524 . . . . 5  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
13 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
1413fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  ( ( x  e.  D  |->  B ) `  A ) )
151adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  A )  ->  A  e.  D )
1613, 15eqeltrd 2357 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  x  e.  D )
17 fvmptdf.2 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
18 eqid 2283 . . . . . . . . . 10  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1918fvmpt2 5608 . . . . . . . . 9  |-  ( ( x  e.  D  /\  B  e.  V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
2016, 17, 19syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  B )
2114, 20eqtr3d 2317 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  A
)  =  B )
2221eqeq2d 2294 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A )  <-> 
( F `  A
)  =  B ) )
23 fvmptdf.3 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  B  ->  ps ) )
2422, 23sylbid 206 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A )  ->  ps ) )
2512, 24syl5 28 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) )
2625ex 423 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) ) )
276, 11, 26exlimd 1803 . 2  |-  ( ph  ->  ( E. x  x  =  A  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) ) )
285, 27mpd 14 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788    e. cmpt 4077   ` cfv 5255
This theorem is referenced by:  fvmptdv  5612  yonedalem4b  14050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263
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