ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funbrfv2b Unicode version

Theorem funbrfv2b 5349
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 5032 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 4670 . . . . 5  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
32ex 113 . . . 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 386 . 2  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  A F B ) ) )
6 funbrfvb 5347 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <-> 
A F B ) )
76pm5.32da 440 . 2  |-  ( Fun 
F  ->  ( ( A  e.  dom  F  /\  ( F `  A )  =  B )  <->  ( A  e.  dom  F  /\  A F B ) ) )
85, 7bitr4d 189 1  |-  ( Fun 
F  ->  ( A F B  <->  ( A  e. 
dom  F  /\  ( F `  A )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   class class class wbr 3845   dom cdm 4438   Rel wrel 4443   Fun wfun 5009   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  brtpos2  6016  xpcomco  6542
  Copyright terms: Public domain W3C validator