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Theorem funbrfv2b 5567
Description: Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
Assertion
Ref Expression
funbrfv2b  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  ( F `  A )  =  B ) ) )

Proof of Theorem funbrfv2b
StepHypRef Expression
1 funrel 5272 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 4911 . . . . 5  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
32ex 423 . . . 4  |-  ( Rel 
F  ->  ( A F B  ->  A  e. 
dom  F ) )
41, 3syl 15 . . 3  |-  ( Fun 
F  ->  ( A F B  ->  A  e. 
dom  F ) )
54pm4.71rd 616 . 2  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  A F B ) ) )
6 funbrfvb 5565 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <-> 
A F B ) )
76pm5.32da 622 . 2  |-  ( Fun 
F  ->  ( ( A  e.  dom  F  /\  ( F `  A )  =  B )  <->  ( A  e.  dom  F  /\  A F B ) ) )
85, 7bitr4d 247 1  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  ( F `  A )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   dom cdm 4689   Rel wrel 4694   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  brtpos2  6240  xpcomco  6952  fseqenlem2  7652  fpwwe2  8265  tayl0  19741  funcnvmptOLD  23234  funcnvmpt  23235
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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