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Theorem f1oeq2 6822
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 6783 . . 3 (𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))
2 foeq2 6802 . . 3 (𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
31, 2anbi12d 631 . 2 (𝐴 = 𝐵 → ((𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶) ↔ (𝐹:𝐵1-1𝐶𝐹:𝐵onto𝐶)))
4 df-f1o 6550 . 2 (𝐹:𝐴1-1-onto𝐶 ↔ (𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶))
5 df-f1o 6550 . 2 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹:𝐵1-1𝐶𝐹:𝐵onto𝐶))
63, 4, 53bitr4g 313 1 (𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  1-1wf1 6540  ontowfo 6541  1-1-ontowf1o 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2724  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550
This theorem is referenced by:  f1oeq23  6824  f1oeq123d  6827  f1oeq2d  6829  resin  6855  isoeq4  7316  breng  8947  brenOLD  8949  f1dmvrnfibi  9335  cnfcom  9694  infxpenc2  10016  fsumf1o  15668  sumsnf  15688  fprodf1o  15889  prodsn  15905  prodsnf  15907  znhash  21113  znunithash  21119  imasf1oxms  23997  wlksnwwlknvbij  29159  clwwlkvbij  29363  eupthp1  29466  derangval  34153  subfacp1lem2a  34166  subfacp1lem3  34168  subfacp1lem5  34170  sumsnd  43700  uspgrsprfo  46516
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