MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmpt3 Unicode version

Theorem fvmpt3 5687
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 2424 . . 3  |-  ( x  =  A  ->  ( B  e.  V  <->  C  e.  V ) )
3 fvmpt3.c . . 3  |-  ( x  e.  D  ->  B  e.  V )
42, 3vtoclga 2925 . 2  |-  ( A  e.  D  ->  C  e.  V )
5 fvmpt3.b . . 3  |-  F  =  ( x  e.  D  |->  B )
61, 5fvmptg 5683 . 2  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
74, 6mpdan 649 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    e. cmpt 4158   ` cfv 5337
This theorem is referenced by:  fvmpt3i  5688  harval  7366  mrcfval  13609  elmptrab  17624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345
  Copyright terms: Public domain W3C validator