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Theorem f1ssr 6557
Description: A function that is one-to-one is also one-to-one on some superset of its range. (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 6552 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
21adantr 483 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹 Fn 𝐴)
3 simpr 487 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → ran 𝐹𝐶)
4 df-f 6335 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
52, 3, 4sylanbrc 585 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴𝐶)
6 df-f1 6336 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
76simprbi 499 . . 3 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
87adantr 483 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → Fun 𝐹)
9 df-f1 6336 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
105, 8, 9sylanbrc 585 1 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wss 3913  ccnv 5530  ran crn 5532  Fun wfun 6325   Fn wfn 6326  wf 6327  1-1wf1 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 209  df-an 399  df-f 6335  df-f1 6336
This theorem is referenced by:  f1resf1  6559  domdifsn  8578  marypha1  8876  m2cpmf1  21327  ausgrusgri  26940  uspgrupgrushgr  26949  usgrumgruspgr  26952  usgruspgrb  26953  usgrres  27077  usgrres1  27084  lindflbs  30949  dimkerim  31034
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