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Theorem f1oeq2 6823
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1oeq2 (𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))

Proof of Theorem f1oeq2
StepHypRef Expression
1 f1eq2 6784 . . 3 (𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))
2 foeq2 6803 . . 3 (𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
31, 2anbi12d 632 . 2 (𝐴 = 𝐵 → ((𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶) ↔ (𝐹:𝐵1-1𝐶𝐹:𝐵onto𝐶)))
4 df-f1o 6551 . 2 (𝐹:𝐴1-1-onto𝐶 ↔ (𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶))
5 df-f1o 6551 . 2 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹:𝐵1-1𝐶𝐹:𝐵onto𝐶))
63, 4, 53bitr4g 314 1 (𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  1-1wf1 6541  ontowfo 6542  1-1-ontowf1o 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551
This theorem is referenced by:  f1oeq23  6825  f1oeq123d  6828  f1oeq2d  6830  resin  6856  isoeq4  7317  breng  8948  brenOLD  8950  f1dmvrnfibi  9336  cnfcom  9695  infxpenc2  10017  fsumf1o  15669  sumsnf  15689  fprodf1o  15890  prodsn  15906  prodsnf  15908  znhash  21114  znunithash  21120  imasf1oxms  23998  wlksnwwlknvbij  29162  clwwlkvbij  29366  eupthp1  29469  derangval  34158  subfacp1lem2a  34171  subfacp1lem3  34173  subfacp1lem5  34175  sumsnd  43710  uspgrsprfo  46526
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