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Theorem fvmpt3 5811
Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
fvmpt3.a  |-  ( x  =  A  ->  B  =  C )
fvmpt3.b  |-  F  =  ( x  e.  D  |->  B )
fvmpt3.c  |-  ( x  e.  D  ->  B  e.  V )
Assertion
Ref Expression
fvmpt3  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Distinct variable groups:    x, A    x, C    x, D    x, V
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmpt3
StepHypRef Expression
1 fvmpt3.a . . . 4  |-  ( x  =  A  ->  B  =  C )
21eleq1d 2504 . . 3  |-  ( x  =  A  ->  ( B  e.  V  <->  C  e.  V ) )
3 fvmpt3.c . . 3  |-  ( x  e.  D  ->  B  e.  V )
42, 3vtoclga 3019 . 2  |-  ( A  e.  D  ->  C  e.  V )
5 fvmpt3.b . . 3  |-  F  =  ( x  e.  D  |->  B )
61, 5fvmptg 5807 . 2  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
74, 6mpdan 651 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    e. cmpt 4269   ` cfv 5457
This theorem is referenced by:  fvmpt3i  5812  harval  7533  mrcfval  13838  elmptrab  17864  wallispi  27808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465
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