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Theorem brfullfun 24558
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 2298 . 2  |-  ( (FullFun
F `  A )  =  B  <->  B  =  (FullFun F `
 A ) )
2 fullfunfnv 24556 . . 3  |- FullFun F  Fn  _V
3 brfullfun.1 . . 3  |-  A  e. 
_V
4 fnbrfvb 5579 . . 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 24557 . . 3  |-  (FullFun F `  A )  =  ( F `  A )
76eqeq2i 2306 . 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 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    Fn wfn 5266   ` cfv 5271  FullFuncfullfn 24464
This theorem is referenced by:  dfrdg4  24560  tfrqfree  24561
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-singleton 24474  df-singles 24475  df-image 24476  df-funpart 24486  df-fullfun 24487
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