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Theorem funbrfv2b 5903
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 5596 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 5226 . . . . 5  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
32ex 434 . . . 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 635 . 2  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  A F B ) ) )
6 funbrfvb 5901 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <-> 
A F B ) )
76pm5.32da 641 . 2  |-  ( Fun 
F  ->  ( ( A  e.  dom  F  /\  ( F `  A )  =  B )  <->  ( A  e.  dom  F  /\  A F B ) ) )
85, 7bitr4d 256 1  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  ( F `  A )  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   class class class wbr 4440   dom cdm 4992   Rel wrel 4997   Fun wfun 5573   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587
This theorem is referenced by:  brtpos2  6951  mpt2curryd  6988  xpcomco  7597  fseqenlem2  8395  fpwwe2  9010  joinfval  15477  joinfval2  15478  meetfval  15491  meetfval2  15492  tayl0  22484  ofpreima  27165  funcnvmptOLD  27167  funcnvmpt  27168  fperdvper  31203
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