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Theorem f1oeq3 5365
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1oeq3 (𝐴 = 𝐵 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))

Proof of Theorem f1oeq3
StepHypRef Expression
1 f1eq3 5332 . . 3 (𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
2 foeq3 5350 . . 3 (𝐴 = 𝐵 → (𝐹:𝐶onto𝐴𝐹:𝐶onto𝐵))
31, 2anbi12d 465 . 2 (𝐴 = 𝐵 → ((𝐹:𝐶1-1𝐴𝐹:𝐶onto𝐴) ↔ (𝐹:𝐶1-1𝐵𝐹:𝐶onto𝐵)))
4 df-f1o 5137 . 2 (𝐹:𝐶1-1-onto𝐴 ↔ (𝐹:𝐶1-1𝐴𝐹:𝐶onto𝐴))
5 df-f1o 5137 . 2 (𝐹:𝐶1-1-onto𝐵 ↔ (𝐹:𝐶1-1𝐵𝐹:𝐶onto𝐵))
63, 4, 53bitr4g 222 1 (𝐴 = 𝐵 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  1-1wf1 5127  ontowfo 5128  1-1-ontowf1o 5129
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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3081  df-ss 3088  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137
This theorem is referenced by:  f1oeq23  5366  f1oeq123d  5369  f1oeq3d  5371  f1ores  5389  resdif  5396  f1osng  5415  f1oresrab  5592  isoeq5  5713  isoini2  5727  mapsnf1o  6638  bren  6648  xpcomf1o  6726  frechashgf1o  10231  sumeq1  11155  fisumss  11192  fsumcnv  11237  prodeq1f  11352  ennnfonelemhf1o  11960  ennnfonelemex  11961
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