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Theorem fvmpt3 5775
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 2478 . . 3  |-  ( x  =  A  ->  ( B  e.  V  <->  C  e.  V ) )
3 fvmpt3.c . . 3  |-  ( x  e.  D  ->  B  e.  V )
42, 3vtoclga 2985 . 2  |-  ( A  e.  D  ->  C  e.  V )
5 fvmpt3.b . . 3  |-  F  =  ( x  e.  D  |->  B )
61, 5fvmptg 5771 . 2  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
74, 6mpdan 650 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    e. cmpt 4234   ` cfv 5421
This theorem is referenced by:  fvmpt3i  5776  harval  7494  mrcfval  13796  elmptrab  17820  wallispi  27694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429
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