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Theorem fvopabf4g 26386
Description: Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
Hypotheses
Ref Expression
fvopabf4g.1  |-  C  e. 
_V
fvopabf4g.2  |-  ( x  =  A  ->  B  =  C )
fvopabf4g.3  |-  F  =  ( x  e.  ( R  ^m  D ) 
|->  B )
Assertion
Ref Expression
fvopabf4g  |-  ( ( D  e.  X  /\  R  e.  Y  /\  A : D --> R )  ->  ( F `  A )  =  C )
Distinct variable groups:    x, A    x, C    x, D    x, R
Allowed substitution hints:    B( x)    F( x)    X( x)    Y( x)

Proof of Theorem fvopabf4g
StepHypRef Expression
1 elmapg 6785 . . . 4  |-  ( ( R  e.  Y  /\  D  e.  X )  ->  ( A  e.  ( R  ^m  D )  <-> 
A : D --> R ) )
21ancoms 439 . . 3  |-  ( ( D  e.  X  /\  R  e.  Y )  ->  ( A  e.  ( R  ^m  D )  <-> 
A : D --> R ) )
32biimp3ar 1282 . 2  |-  ( ( D  e.  X  /\  R  e.  Y  /\  A : D --> R )  ->  A  e.  ( R  ^m  D ) )
4 fvopabf4g.2 . . 3  |-  ( x  =  A  ->  B  =  C )
5 fvopabf4g.3 . . 3  |-  F  =  ( x  e.  ( R  ^m  D ) 
|->  B )
6 fvopabf4g.1 . . 3  |-  C  e. 
_V
74, 5, 6fvmpt 5602 . 2  |-  ( A  e.  ( R  ^m  D )  ->  ( F `  A )  =  C )
83, 7syl 15 1  |-  ( ( D  e.  X  /\  R  e.  Y  /\  A : D --> R )  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774
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