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Theorem fneqeql 5572
 Description: Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
fneqeql

Proof of Theorem fneqeql
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqfnfv 5562 . . 3
2 eqcom 2159 . . . 4
3 rabid2 2633 . . . 4
42, 3bitri 183 . . 3
51, 4bitr4di 197 . 2
6 fndmin 5571 . . 3
76eqeq1d 2166 . 2
85, 7bitr4d 190 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1335  wral 2435  crab 2439   cin 3101   cdm 4583   wfn 5162  cfv 5167 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-iota 5132  df-fun 5169  df-fn 5170  df-fv 5175 This theorem is referenced by:  fneqeql2  5573  fnreseql  5574
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