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

Theorem fnbrfvb 5732
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 2443 . . . 4  |-  ( F `
 B )  =  ( F `  B
)
2 fvex 5701 . . . . 5  |-  ( F `
 B )  e. 
_V
3 eqeq2 2452 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  (
( F `  B
)  =  x  <->  ( F `  B )  =  ( F `  B ) ) )
4 breq2 4296 . . . . . . 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 5515 . . . . . 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 3024 . . . 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 4296 . . 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 5508 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
15 funbrfv 5730 . . . 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 1369    e. wcel 1756   E!weu 2253   class class class wbr 4292   Fun wfun 5412    Fn wfn 5413   ` cfv 5418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426
This theorem is referenced by:  fnopfvb  5733  funbrfvb  5734  dffn5  5737  fnsnfv  5751  fndmdif  5807  dffo4  5859  dff13  5971  isomin  6028  isoini  6029  1stconst  6661  2ndconst  6662  fsplit  6677  seqomlem3  6907  seqomlem4  6908  nqerrel  9101  imasleval  14479  znleval  17987  axcontlem5  23214  elnlfn  25332  adjbd1o  25489  feqmptdf  25978  br1steq  27585  br2ndeq  27586  trpredpred  27692  fvbigcup  27933  fvsingle  27951  imageval  27961  brfullfun  27979  pw2f1ocnv  29386  funressnfv  30034  fnbrafvb  30060
  Copyright terms: Public domain W3C validator