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Theorem funbrafv2b 27685
Description: Function value in terms of a binary relation, analogous to funbrfv2b 5703. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
funbrafv2b  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  ( F''' A )  =  B ) ) )

Proof of Theorem funbrafv2b
StepHypRef Expression
1 funrel 5404 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 5035 . . . . 5  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
32ex 424 . . . 4  |-  ( Rel 
F  ->  ( A F B  ->  A  e. 
dom  F ) )
41, 3syl 16 . . 3  |-  ( Fun 
F  ->  ( A F B  ->  A  e. 
dom  F ) )
54pm4.71rd 617 . 2  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  A F B ) ) )
6 funbrafvb 27682 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F''' A )  =  B  <->  A F B ) )
76pm5.32da 623 . 2  |-  ( Fun 
F  ->  ( ( A  e.  dom  F  /\  ( F''' A )  =  B )  <->  ( A  e. 
dom  F  /\  A F B ) ) )
85, 7bitr4d 248 1  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  ( F''' A )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4146   dom cdm 4811   Rel wrel 4816   Fun wfun 5381  '''cafv 27633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-res 4823  df-iota 5351  df-fun 5389  df-fn 5390  df-fv 5395  df-dfat 27635  df-afv 27636
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