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Theorem funbrafv 27692
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 5705. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
funbrafv  |-  ( Fun 
F  ->  ( A F B  ->  ( F''' A )  =  B ) )

Proof of Theorem funbrafv
StepHypRef Expression
1 funrel 5412 . . 3  |-  ( Fun 
F  ->  Rel  F )
2 releldm 5043 . . . . . . . 8  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
3 funbrafvb 27690 . . . . . . . . . 10  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F''' A )  =  B  <->  A F B ) )
43biimprd 215 . . . . . . . . 9  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A F B  ->  ( F''' A )  =  B ) )
54expcom 425 . . . . . . . 8  |-  ( A  e.  dom  F  -> 
( Fun  F  ->  ( A F B  -> 
( F''' A )  =  B ) ) )
62, 5syl 16 . . . . . . 7  |-  ( ( Rel  F  /\  A F B )  ->  ( Fun  F  ->  ( A F B  ->  ( F''' A )  =  B ) ) )
76ex 424 . . . . . 6  |-  ( Rel 
F  ->  ( A F B  ->  ( Fun 
F  ->  ( A F B  ->  ( F''' A )  =  B ) ) ) )
87com14 84 . . . . 5  |-  ( A F B  ->  ( A F B  ->  ( Fun  F  ->  ( Rel  F  ->  ( F''' A )  =  B ) ) ) )
98pm2.43i 45 . . . 4  |-  ( A F B  ->  ( Fun  F  ->  ( Rel  F  ->  ( F''' A )  =  B ) ) )
109com13 76 . . 3  |-  ( Rel 
F  ->  ( Fun  F  ->  ( A F B  ->  ( F''' A )  =  B ) ) )
111, 10syl 16 . 2  |-  ( Fun 
F  ->  ( Fun  F  ->  ( A F B  ->  ( F''' A )  =  B ) ) )
1211pm2.43i 45 1  |-  ( Fun 
F  ->  ( A F B  ->  ( F''' A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4154   dom cdm 4819   Rel wrel 4824   Fun wfun 5389  '''cafv 27641
This theorem is referenced by:  afvelima  27701
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 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-res 4831  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403  df-dfat 27643  df-afv 27644
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