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Theorem funbrafv 28126
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 5577. (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 5288 . . 3  |-  ( Fun 
F  ->  Rel  F )
2 releldm 4927 . . . . . . . 8  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
3 funbrafvb 28124 . . . . . . . . . 10  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F''' A )  =  B  <->  A F B ) )
43biimprd 214 . . . . . . . . 9  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A F B  ->  ( F''' A )  =  B ) )
54expcom 424 . . . . . . . 8  |-  ( A  e.  dom  F  -> 
( Fun  F  ->  ( A F B  -> 
( F''' A )  =  B ) ) )
62, 5syl 15 . . . . . . 7  |-  ( ( Rel  F  /\  A F B )  ->  ( Fun  F  ->  ( A F B  ->  ( F''' A )  =  B ) ) )
76ex 423 . . . . . 6  |-  ( Rel 
F  ->  ( A F B  ->  ( Fun 
F  ->  ( A F B  ->  ( F''' A )  =  B ) ) ) )
87com14 82 . . . . 5  |-  ( A F B  ->  ( A F B  ->  ( Fun  F  ->  ( Rel  F  ->  ( F''' A )  =  B ) ) ) )
98pm2.43i 43 . . . 4  |-  ( A F B  ->  ( Fun  F  ->  ( Rel  F  ->  ( F''' A )  =  B ) ) )
109com13 74 . . 3  |-  ( Rel 
F  ->  ( Fun  F  ->  ( A F B  ->  ( F''' A )  =  B ) ) )
111, 10syl 15 . 2  |-  ( Fun 
F  ->  ( Fun  F  ->  ( A F B  ->  ( F''' A )  =  B ) ) )
1211pm2.43i 43 1  |-  ( Fun 
F  ->  ( A F B  ->  ( F''' A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   dom cdm 4705   Rel wrel 4710   Fun wfun 5265  '''cafv 28075
This theorem is referenced by:  afvelima  28135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-dfat 28077  df-afv 28078
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