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Theorem f1ssr 6145
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 6140 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
21adantr 480 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹 Fn 𝐴)
3 simpr 476 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → ran 𝐹𝐶)
4 df-f 5930 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
52, 3, 4sylanbrc 699 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴𝐶)
6 df-f1 5931 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
76simprbi 479 . . 3 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
87adantr 480 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → Fun 𝐹)
9 df-f1 5931 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
105, 8, 9sylanbrc 699 1 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wss 3607  ccnv 5142  ran crn 5144  Fun wfun 5920   Fn wfn 5921  wf 5922  1-1wf1 5923
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 197  df-an 385  df-f 5930  df-f1 5931
This theorem is referenced by:  domdifsn  8084  marypha1  8381  m2cpmf1  20596  ausgrusgri  26108  uspgrupgrushgr  26117  usgrumgruspgr  26120  usgruspgrb  26121  usgrres  26245  usgrres1  26252
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