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Theorem fnbrfvb 5726
Description: Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fnbrfvb  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )

Proof of Theorem fnbrfvb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . 4  |-  ( F `
 B )  =  ( F `  B
)
2 fvex 5701 . . . . 5  |-  ( F `
 B )  e. 
_V
3 eqeq2 2413 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  (
( F `  B
)  =  x  <->  ( F `  B )  =  ( F `  B ) ) )
4 breq2 4176 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  ( B F x  <->  B F
( F `  B
) ) )
53, 4bibi12d 313 . . . . . 6  |-  ( x  =  ( F `  B )  ->  (
( ( F `  B )  =  x  <-> 
B F x )  <-> 
( ( F `  B )  =  ( F `  B )  <-> 
B F ( F `
 B ) ) ) )
65imbi2d 308 . . . . 5  |-  ( x  =  ( F `  B )  ->  (
( ( F  Fn  A  /\  B  e.  A
)  ->  ( ( F `  B )  =  x  <->  B F x ) )  <->  ( ( F  Fn  A  /\  B  e.  A )  ->  (
( F `  B
)  =  ( F `
 B )  <->  B F
( F `  B
) ) ) ) )
7 fneu 5508 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! x  B F x )
8 tz6.12c 5709 . . . . . 6  |-  ( E! x  B F x  ->  ( ( F `
 B )  =  x  <->  B F x ) )
97, 8syl 16 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  x  <-> 
B F x ) )
102, 6, 9vtocl 2966 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  ( F `  B )  <-> 
B F ( F `
 B ) ) )
111, 10mpbii 203 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  B F ( F `
 B ) )
12 breq2 4176 . . 3  |-  ( ( F `  B )  =  C  ->  ( B F ( F `  B )  <->  B F C ) )
1311, 12syl5ibcom 212 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  ->  B F C ) )
14 fnfun 5501 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
15 funbrfv 5724 . . . 4  |-  ( Fun 
F  ->  ( B F C  ->  ( F `
 B )  =  C ) )
1614, 15syl 16 . . 3  |-  ( F  Fn  A  ->  ( B F C  ->  ( F `  B )  =  C ) )
1716adantr 452 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( B F C  ->  ( F `  B )  =  C ) )
1813, 17impbid 184 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E!weu 2254   class class class wbr 4172   Fun wfun 5407    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  fnopfvb  5727  funbrfvb  5728  dffn5  5731  fnsnfv  5745  fndmdif  5793  dffo4  5844  dff13  5963  isomin  6016  isoini  6017  1stconst  6394  2ndconst  6395  fsplit  6410  seqomlem3  6668  seqomlem4  6669  nqerrel  8765  imasleval  13721  znleval  16790  elnlfn  23384  adjbd1o  23541  feqmptdf  24028  br1steq  25344  br2ndeq  25345  trpredpred  25445  fvbigcup  25656  fvsingle  25673  imageval  25683  brfullfun  25701  axcontlem5  25811  pw2f1ocnv  26998  funressnfv  27859  fnbrafvb  27885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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