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Theorem f1oeq23 5148
Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
Assertion
Ref Expression
f1oeq23 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))

Proof of Theorem f1oeq23
StepHypRef Expression
1 f1oeq2 5146 . 2 (𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
2 f1oeq3 5147 . 2 (𝐶 = 𝐷 → (𝐹:𝐵1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))
31, 2sylan9bb 443 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  1-1-ontowf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by: (None)
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