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Theorem fnbrfvb 5157
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  F `  C  F C

Proof of Theorem fnbrfvb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqid 2037 . . . 4  F `
 F `
2 funfvex 5135 . . . . . 6  Fun  F  dom  F  F `  _V
32funfni 4942 . . . . 5  F  Fn  F `  _V
4 eqeq2 2046 . . . . . . . 8  F `  F `  F `
 F `
5 breq2 3759 . . . . . . . 8  F `  F  F F `
64, 5bibi12d 224 . . . . . . 7  F `  F `  F  F `  F `  F F `
76imbi2d 219 . . . . . 6  F `  F  Fn  F `  F  F  Fn  F `
 F `  F F `
8 fneu 4946 . . . . . . 7  F  Fn  F
9 tz6.12c 5146 . . . . . . 7  F  F `  F
108, 9syl 14 . . . . . 6  F  Fn  F `  F
117, 10vtoclg 2607 . . . . 5  F `  _V  F  Fn  F `  F `  F F `
123, 11mpcom 32 . . . 4  F  Fn  F `  F `  F F `
131, 12mpbii 136 . . 3  F  Fn  F F `
14 breq2 3759 . . 3  F `  C  F F `  F C
1513, 14syl5ibcom 144 . 2  F  Fn  F `  C  F C
16 fnfun 4939 . . . 4  F  Fn  Fun  F
17 funbrfv 5155 . . . 4  Fun 
F  F C  F `  C
1816, 17syl 14 . . 3  F  Fn  F C  F `  C
1918adantr 261 . 2  F  Fn  F C  F `  C
2015, 19impbid 120 1  F  Fn  F `  C  F C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  weu 1897   _Vcvv 2551   class class class wbr 3755   Fun wfun 4839    Fn wfn 4840   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fnopfvb  5158  funbrfvb  5159  dffn5im  5162  fnsnfv  5175  fndmdif  5215  dffo4  5258  dff13  5350  isoini  5400  1stconst  5784  2ndconst  5785
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