MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnbrfvb Structured version   Unicode version

Theorem fnbrfvb 5913
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 2457 . . . 4  |-  ( F `
 B )  =  ( F `  B
)
2 fvex 5882 . . . . 5  |-  ( F `
 B )  e. 
_V
3 eqeq2 2472 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  (
( F `  B
)  =  x  <->  ( F `  B )  =  ( F `  B ) ) )
4 breq2 4460 . . . . . . 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 5691 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! x  B F x )
8 tz6.12c 5891 . . . . . 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 3161 . . . 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 4460 . . 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 5684 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
15 funbrfv 5911 . . . 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 1395    e. wcel 1819   E!weu 2283   class class class wbr 4456   Fun wfun 5588    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  fnopfvb  5914  funbrfvb  5915  dffn5  5918  fnsnfv  5933  fndmdif  5992  dffo4  6048  dff13  6167  isomin  6234  isoini  6235  1stconst  6887  2ndconst  6888  fsplit  6904  seqomlem3  7135  seqomlem4  7136  nqerrel  9327  imasleval  14958  znleval  18720  axcontlem5  24398  elnlfn  26974  adjbd1o  27131  fcoinvbr  27603  feqmptdf  27650  br1steq  29421  br2ndeq  29422  trpredpred  29528  fvbigcup  29757  fvsingle  29775  imageval  29785  brfullfun  29803  pw2f1ocnv  31183  funressnfv  32416  fnbrafvb  32442
  Copyright terms: Public domain W3C validator