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Theorem fnbrfvb 5919
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 2423 . . . 4  |-  ( F `
 B )  =  ( F `  B
)
2 fvex 5889 . . . . 5  |-  ( F `
 B )  e. 
_V
3 eqeq2 2438 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  (
( F `  B
)  =  x  <->  ( F `  B )  =  ( F `  B ) ) )
4 breq2 4425 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  ( B F x  <->  B F
( F `  B
) ) )
53, 4bibi12d 323 . . . . . 6  |-  ( x  =  ( F `  B )  ->  (
( ( F `  B )  =  x  <-> 
B F x )  <-> 
( ( F `  B )  =  ( F `  B )  <-> 
B F ( F `
 B ) ) ) )
65imbi2d 318 . . . . 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 5696 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! x  B F x )
8 tz6.12c 5898 . . . . . 6  |-  ( E! x  B F x  ->  ( ( F `
 B )  =  x  <->  B F x ) )
97, 8syl 17 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  x  <-> 
B F x ) )
102, 6, 9vtocl 3134 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  ( F `  B )  <-> 
B F ( F `
 B ) ) )
111, 10mpbii 215 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  B F ( F `
 B ) )
12 breq2 4425 . . 3  |-  ( ( F `  B )  =  C  ->  ( B F ( F `  B )  <->  B F C ) )
1311, 12syl5ibcom 224 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  ->  B F C ) )
14 fnfun 5689 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
15 funbrfv 5917 . . . 4  |-  ( Fun 
F  ->  ( B F C  ->  ( F `
 B )  =  C ) )
1614, 15syl 17 . . 3  |-  ( F  Fn  A  ->  ( B F C  ->  ( F `  B )  =  C ) )
1716adantr 467 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( B F C  ->  ( F `  B )  =  C ) )
1813, 17impbid 194 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   E!weu 2266   class class class wbr 4421   Fun wfun 5593    Fn wfn 5594   ` cfv 5599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fn 5602  df-fv 5607
This theorem is referenced by:  fnopfvb  5920  funbrfvb  5921  dffn5  5924  fnsnfv  5939  fndmdif  5999  dffo4  6051  dff13  6172  isomin  6241  isoini  6242  1stconst  6893  2ndconst  6894  fsplit  6910  seqomlem3  7175  seqomlem4  7176  nqerrel  9359  imasleval  15440  znleval  19117  axcontlem5  24990  elnlfn  27573  adjbd1o  27730  fcoinvbr  28211  feqmptdf  28254  br1steq  30415  br2ndeq  30416  fv1stcnv  30423  fv2ndcnv  30424  trpredpred  30470  fvbigcup  30668  fvsingle  30686  imageval  30696  brfullfun  30714  poimirlem2  31862  poimirlem23  31883  pw2f1ocnv  35818  funressnfv  38348  fnbrafvb  38374
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