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Theorem fnbrfvb 5290
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 2083 . . . 4  |-  ( F `
 B )  =  ( F `  B
)
2 funfvex 5267 . . . . . 6  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( F `  B
)  e.  _V )
32funfni 5067 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F `  B
)  e.  _V )
4 eqeq2 2092 . . . . . . . 8  |-  ( x  =  ( F `  B )  ->  (
( F `  B
)  =  x  <->  ( F `  B )  =  ( F `  B ) ) )
5 breq2 3815 . . . . . . . 8  |-  ( x  =  ( F `  B )  ->  ( B F x  <->  B F
( F `  B
) ) )
64, 5bibi12d 233 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  (
( ( F `  B )  =  x  <-> 
B F x )  <-> 
( ( F `  B )  =  ( F `  B )  <-> 
B F ( F `
 B ) ) ) )
76imbi2d 228 . . . . . 6  |-  ( 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
) ) ) ) )
8 fneu 5071 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! x  B F x )
9 tz6.12c 5279 . . . . . . 7  |-  ( E! x  B F x  ->  ( ( F `
 B )  =  x  <->  B F x ) )
108, 9syl 14 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  x  <-> 
B F x ) )
117, 10vtoclg 2669 . . . . 5  |-  ( ( F `  B )  e.  _V  ->  (
( F  Fn  A  /\  B  e.  A
)  ->  ( ( F `  B )  =  ( F `  B )  <->  B F
( F `  B
) ) ) )
123, 11mpcom 36 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  ( F `  B )  <-> 
B F ( F `
 B ) ) )
131, 12mpbii 146 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  B F ( F `
 B ) )
14 breq2 3815 . . 3  |-  ( ( F `  B )  =  C  ->  ( B F ( F `  B )  <->  B F C ) )
1513, 14syl5ibcom 153 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  ->  B F C ) )
16 fnfun 5064 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
17 funbrfv 5288 . . . 4  |-  ( Fun 
F  ->  ( B F C  ->  ( F `
 B )  =  C ) )
1816, 17syl 14 . . 3  |-  ( F  Fn  A  ->  ( B F C  ->  ( F `  B )  =  C ) )
1918adantr 270 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( B F C  ->  ( F `  B )  =  C ) )
2015, 19impbid 127 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   E!weu 1943   _Vcvv 2612   class class class wbr 3811   Fun wfun 4963    Fn wfn 4964   ` cfv 4969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fn 4972  df-fv 4977
This theorem is referenced by:  fnopfvb  5291  funbrfvb  5292  dffn5im  5295  fnsnfv  5308  fndmdif  5349  dffo4  5392  dff13  5487  isoini  5536  1stconst  5921  2ndconst  5922
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