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Theorem f1ssr 5335
 Description: Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Assertion
Ref Expression
f1ssr ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)

Proof of Theorem f1ssr
StepHypRef Expression
1 f1fn 5330 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
21adantr 274 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹 Fn 𝐴)
3 simpr 109 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → ran 𝐹𝐶)
4 df-f 5127 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
52, 3, 4sylanbrc 413 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴𝐶)
6 df-f1 5128 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
76simprbi 273 . . 3 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
87adantr 274 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → Fun 𝐹)
9 df-f1 5128 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
105, 8, 9sylanbrc 413 1 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ⊆ wss 3071  ◡ccnv 4538  ran crn 4540  Fun wfun 5117   Fn wfn 5118  ⟶wf 5119  –1-1→wf1 5120 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116  df-f 5127  df-f1 5128 This theorem is referenced by:  f1ff1  5336  difinfsn  6985
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