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Theorem fvmptdf 5627
Description: Alternate deduction version of fvmpt 5618, 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 2809 . . . 4  |-  ( A  e.  D  ->  A  e.  _V )
31, 2syl 15 . . 3  |-  ( ph  ->  A  e.  _V )
4 isset 2805 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
53, 4sylib 188 . 2  |-  ( ph  ->  E. x  x  =  A )
6 nfv 1609 . . 3  |-  F/ x ph
7 fvmptdf.4 . . . . 5  |-  F/_ x F
8 nfmpt1 4125 . . . . 5  |-  F/_ x
( x  e.  D  |->  B )
97, 8nfeq 2439 . . . 4  |-  F/ x  F  =  ( x  e.  D  |->  B )
10 fvmptdf.5 . . . 4  |-  F/ x ps
119, 10nfim 1781 . . 3  |-  F/ x
( F  =  ( x  e.  D  |->  B )  ->  ps )
12 fveq1 5540 . . . . 5  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
13 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
1413fveq2d 5545 . . . . . . . 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 2370 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  x  e.  D )
17 fvmptdf.2 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
18 eqid 2296 . . . . . . . . . 10  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1918fvmpt2 5624 . . . . . . . . 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 2330 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  A
)  =  B )
2221eqeq2d 2307 . . . . . 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 1815 . 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 1531   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419   _Vcvv 2801    e. cmpt 4093   ` cfv 5271
This theorem is referenced by:  fvmptdv  5628  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-13 1698  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-pow 4204  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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279
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