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Theorem funbrfv2b 5246
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 4947 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 4597 . . . . 5  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
32ex 112 . . . 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 380 . 2  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  A F B ) ) )
6 funbrfvb 5244 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <-> 
A F B ) )
76pm5.32da 433 . 2  |-  ( Fun 
F  ->  ( ( A  e.  dom  F  /\  ( F `  A )  =  B )  <->  ( A  e.  dom  F  /\  A F B ) ) )
85, 7bitr4d 184 1  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  ( F `  A )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   class class class wbr 3792   dom cdm 4373   Rel wrel 4378   Fun wfun 4924   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by:  brtpos2  5897  xpcomco  6331
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