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Theorem fnbrfvb 5908
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 2467 . . . 4  |-  ( F `
 B )  =  ( F `  B
)
2 fvex 5876 . . . . 5  |-  ( F `
 B )  e. 
_V
3 eqeq2 2482 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  (
( F `  B
)  =  x  <->  ( F `  B )  =  ( F `  B ) ) )
4 breq2 4451 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  ( B F x  <->  B F
( F `  B
) ) )
53, 4bibi12d 321 . . . . . 6  |-  ( x  =  ( F `  B )  ->  (
( ( F `  B )  =  x  <-> 
B F x )  <-> 
( ( F `  B )  =  ( F `  B )  <-> 
B F ( F `
 B ) ) ) )
65imbi2d 316 . . . . 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 5685 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! x  B F x )
8 tz6.12c 5885 . . . . . 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 3165 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  ( F `  B )  <-> 
B F ( F `
 B ) ) )
111, 10mpbii 211 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  B F ( F `
 B ) )
12 breq2 4451 . . 3  |-  ( ( F `  B )  =  C  ->  ( B F ( F `  B )  <->  B F C ) )
1311, 12syl5ibcom 220 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  ->  B F C ) )
14 fnfun 5678 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
15 funbrfv 5906 . . . 4  |-  ( Fun 
F  ->  ( B F C  ->  ( F `
 B )  =  C ) )
1614, 15syl 16 . . 3  |-  ( F  Fn  A  ->  ( B F C  ->  ( F `  B )  =  C ) )
1716adantr 465 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( B F C  ->  ( F `  B )  =  C ) )
1813, 17impbid 191 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E!weu 2275   class class class wbr 4447   Fun wfun 5582    Fn wfn 5583   ` cfv 5588
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 5551  df-fun 5590  df-fn 5591  df-fv 5596
This theorem is referenced by:  fnopfvb  5909  funbrfvb  5910  dffn5  5913  fnsnfv  5927  fndmdif  5985  dffo4  6037  dff13  6154  isomin  6221  isoini  6222  1stconst  6871  2ndconst  6872  fsplit  6888  seqomlem3  7117  seqomlem4  7118  nqerrel  9310  imasleval  14796  znleval  18388  axcontlem5  23975  elnlfn  26551  adjbd1o  26708  fcoinvbr  27162  feqmptdf  27201  br1steq  28809  br2ndeq  28810  trpredpred  28916  fvbigcup  29157  fvsingle  29175  imageval  29185  brfullfun  29203  pw2f1ocnv  30611  funressnfv  31708  fnbrafvb  31734
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