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Theorem fvmpt3 5954
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 2536 . . 3  |-  ( x  =  A  ->  ( B  e.  V  <->  C  e.  V ) )
3 fvmpt3.c . . 3  |-  ( x  e.  D  ->  B  e.  V )
42, 3vtoclga 3177 . 2  |-  ( A  e.  D  ->  C  e.  V )
5 fvmpt3.b . . 3  |-  F  =  ( x  e.  D  |->  B )
61, 5fvmptg 5949 . 2  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
74, 6mpdan 668 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    |-> 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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  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-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-iota 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  fvmpt3i  5955  harval  7989  mrcfval  14866  elmptrab  20155  sgnsv  27476  wallispi  31597
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