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Theorem funbrfv2b 5512
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 5186 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 4820 . . . . 5  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
32ex 114 . . . 4  |-  ( Rel 
F  ->  ( A F B  ->  A  e. 
dom  F ) )
41, 3syl 14 . . 3  |-  ( Fun 
F  ->  ( A F B  ->  A  e. 
dom  F ) )
54pm4.71rd 392 . 2  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  A F B ) ) )
6 funbrfvb 5510 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <-> 
A F B ) )
76pm5.32da 448 . 2  |-  ( Fun 
F  ->  ( ( A  e.  dom  F  /\  ( F `  A )  =  B )  <->  ( A  e.  dom  F  /\  A F B ) ) )
85, 7bitr4d 190 1  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  ( F `  A )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   class class class wbr 3965   dom cdm 4585   Rel wrel 4590   Fun wfun 5163   ` cfv 5169
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-iota 5134  df-fun 5171  df-fn 5172  df-fv 5177
This theorem is referenced by:  brtpos2  6195  xpcomco  6768
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