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Theorem fvmptdf 5961
Description: Alternate deduction version of fvmpt 5950, 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 1683 . 2  |-  F/ x ph
2 fvmptdf.4 . . . 4  |-  F/_ x F
3 nfmpt1 4536 . . . 4  |-  F/_ x
( x  e.  D  |->  B )
42, 3nfeq 2640 . . 3  |-  F/ x  F  =  ( x  e.  D  |->  B )
5 fvmptdf.5 . . 3  |-  F/ x ps
64, 5nfim 1867 . 2  |-  F/ x
( F  =  ( x  e.  D  |->  B )  ->  ps )
7 fvmptdf.1 . . . 4  |-  ( ph  ->  A  e.  D )
8 elex 3122 . . . 4  |-  ( A  e.  D  ->  A  e.  _V )
97, 8syl 16 . . 3  |-  ( ph  ->  A  e.  _V )
10 isset 3117 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
119, 10sylib 196 . 2  |-  ( ph  ->  E. x  x  =  A )
12 fveq1 5865 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
13 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
1413fveq2d 5870 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  ( ( x  e.  D  |->  B ) `  A ) )
157adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  =  A )  ->  A  e.  D )
1613, 15eqeltrd 2555 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  e.  D )
17 fvmptdf.2 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
18 eqid 2467 . . . . . . . 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 661 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  B )
2114, 20eqtr3d 2510 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  A
)  =  B )
2221eqeq2d 2481 . . . 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 215 . . 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 1929 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596   F/wnf 1599    e. wcel 1767   F/_wnfc 2615   _Vcvv 3113    |-> cmpt 4505   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  fvmptdv  5962  yonedalem4b  15403
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