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Theorem funfvbrb 5992
Description: Two ways to say that  A is in the domain of  F. (Contributed by Mario Carneiro, 1-May-2014.)
Assertion
Ref Expression
funfvbrb  |-  ( Fun 
F  ->  ( A  e.  dom  F  <->  A F
( F `  A
) ) )

Proof of Theorem funfvbrb
StepHypRef Expression
1 funfvop 5991 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 df-br 4448 . . 3  |-  ( A F ( F `  A )  <->  <. A , 
( F `  A
) >.  e.  F )
31, 2sylibr 212 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A F ( F `  A ) )
4 funrel 5603 . . 3  |-  ( Fun 
F  ->  Rel  F )
5 releldm 5233 . . 3  |-  ( ( Rel  F  /\  A F ( F `  A ) )  ->  A  e.  dom  F )
64, 5sylan 471 . 2  |-  ( ( Fun  F  /\  A F ( F `  A ) )  ->  A  e.  dom  F )
73, 6impbida 830 1  |-  ( Fun 
F  ->  ( A  e.  dom  F  <->  A F
( F `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   <.cop 4033   class class class wbr 4447   dom cdm 4999   Rel wrel 5004   Fun wfun 5580   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  fmptco  6052  fpwwe2lem13  9016  fpwwe2  9017  climdm  13333  invco  15019  funciso  15094  ffthiso  15149  fuciso  15195  setciso  15269  catciso  15285  lmcau  21483  dvcnp  22054  dvadd  22075  dvmul  22076  dvaddf  22077  dvmulf  22078  dvco  22082  dvcof  22083  dvcjbr  22084  dvcnvlem  22109  dvferm1  22118  dvferm2  22120  ulmdm  22519  ulmdvlem3  22528  minvecolem4a  25466  hlimf  25828  hhsscms  25868  occllem  25894  occl  25895  chscllem4  26231  fmptcof2  27171  heiborlem9  29916  bfplem1  29919
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