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Theorem f1ff1 5224
Description: If a function is one-to-one from A to B and is also a function from A to C, then it is a one-to-one function from A to C. (Contributed by BJ, 4-Jul-2022.)
Assertion
Ref Expression
f1ff1 ((𝐹:𝐴1-1𝐵𝐹:𝐴𝐶) → 𝐹:𝐴1-1𝐶)

Proof of Theorem f1ff1
StepHypRef Expression
1 frn 5169 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
2 f1ssr 5223 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)
31, 2sylan2 280 1 ((𝐹:𝐴1-1𝐵𝐹:𝐴𝐶) → 𝐹:𝐴1-1𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wss 2999  ran crn 4439  wf 5011  1-1wf1 5012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-f 5019  df-f1 5020
This theorem is referenced by:  f1resf1  5226  inresflem  6750
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