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Theorem brfullfun 24486
Description: A binary relationship form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brfullfun.1  |-  A  e. 
_V
brfullfun.2  |-  B  e. 
_V
Assertion
Ref Expression
brfullfun  |-  ( AFullFun
F B  <->  B  =  ( F `  A ) )

Proof of Theorem brfullfun
StepHypRef Expression
1 eqcom 2285 . 2  |-  ( (FullFun
F `  A )  =  B  <->  B  =  (FullFun F `
 A ) )
2 fullfunfnv 24484 . . 3  |- FullFun F  Fn  _V
3 brfullfun.1 . . 3  |-  A  e. 
_V
4 fnbrfvb 5563 . . 3  |-  ( (FullFun
F  Fn  _V  /\  A  e.  _V )  ->  ( (FullFun F `  A )  =  B  <-> 
AFullFun F B ) )
52, 3, 4mp2an 653 . 2  |-  ( (FullFun
F `  A )  =  B  <->  AFullFun F B )
6 fullfunfv 24485 . . 3  |-  (FullFun F `  A )  =  ( F `  A )
76eqeq2i 2293 . 2  |-  ( B  =  (FullFun F `  A )  <->  B  =  ( F `  A ) )
81, 5, 73bitr3i 266 1  |-  ( AFullFun
F B  <->  B  =  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    Fn wfn 5250   ` cfv 5255  FullFuncfullfn 24393
This theorem is referenced by:  dfrdg4  24488  tfrqfree  24489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-singleton 24403  df-singles 24404  df-image 24405  df-funpart 24415  df-fullfun 24416
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