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Theorem fvmpt3 5468
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 2186 . . 3  |-  ( x  =  A  ->  ( B  e.  V  <->  C  e.  V ) )
3 fvmpt3.c . . 3  |-  ( x  e.  D  ->  B  e.  V )
42, 3vtoclga 2726 . 2  |-  ( A  e.  D  ->  C  e.  V )
5 fvmpt3.b . . 3  |-  F  =  ( x  e.  D  |->  B )
61, 5fvmptg 5465 . 2  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
74, 6mpdan 417 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465    |-> cmpt 3959   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101
This theorem is referenced by:  fvmpt3i  5469  frec2uzsucd  10142
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