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Theorem f1oeq3 5328
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 5295 . . 3 (𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
2 foeq3 5313 . . 3 (𝐴 = 𝐵 → (𝐹:𝐶onto𝐴𝐹:𝐶onto𝐵))
31, 2anbi12d 464 . 2 (𝐴 = 𝐵 → ((𝐹:𝐶1-1𝐴𝐹:𝐶onto𝐴) ↔ (𝐹:𝐶1-1𝐵𝐹:𝐶onto𝐵)))
4 df-f1o 5100 . 2 (𝐹:𝐶1-1-onto𝐴 ↔ (𝐹:𝐶1-1𝐴𝐹:𝐶onto𝐴))
5 df-f1o 5100 . 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 1316  1-1wf1 5090  ontowfo 5091  1-1-ontowf1o 5092
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100
This theorem is referenced by:  f1oeq23  5329  f1oeq123d  5332  f1ores  5350  resdif  5357  f1osng  5376  f1oresrab  5553  isoeq5  5674  isoini2  5688  mapsnf1o  6599  bren  6609  xpcomf1o  6687  frechashgf1o  10169  sumeq1  11092  fisumss  11129  fsumcnv  11174  ennnfonelemhf1o  11853  ennnfonelemex  11854
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