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Theorem f1oeq2 6774
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 6735 . . 3 (𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))
2 foeq2 6754 . . 3 (𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
31, 2anbi12d 632 . 2 (𝐴 = 𝐵 → ((𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶) ↔ (𝐹:𝐵1-1𝐶𝐹:𝐵onto𝐶)))
4 df-f1o 6504 . 2 (𝐹:𝐴1-1-onto𝐶 ↔ (𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶))
5 df-f1o 6504 . 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 6494  ontowfo 6495  1-1-ontowf1o 6496
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 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2729  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  f1oeq23  6776  f1oeq123d  6779  f1oeq2d  6781  resin  6807  isoeq4  7266  breng  8893  brenOLD  8895  f1dmvrnfibi  9281  cnfcom  9637  infxpenc2  9959  fsumf1o  15609  sumsnf  15629  fprodf1o  15830  prodsn  15846  prodsnf  15848  znhash  20968  znunithash  20974  imasf1oxms  23848  wlksnwwlknvbij  28856  clwwlkvbij  29060  eupthp1  29163  derangval  33764  subfacp1lem2a  33777  subfacp1lem3  33779  subfacp1lem5  33781  sumsnd  43238  uspgrsprfo  46057
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