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Theorem f1ssr 6790
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 6784 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
21adantr 482 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹 Fn 𝐴)
3 simpr 486 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → ran 𝐹𝐶)
4 df-f 6543 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
52, 3, 4sylanbrc 584 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴𝐶)
6 df-f1 6544 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
76simprbi 498 . . 3 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
87adantr 482 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → Fun 𝐹)
9 df-f1 6544 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
105, 8, 9sylanbrc 584 1 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wss 3946  ccnv 5673  ran crn 5675  Fun wfun 6533   Fn wfn 6534  wf 6535  1-1wf1 6536
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 206  df-an 398  df-f 6543  df-f1 6544
This theorem is referenced by:  f1resf1  6792  domdifsn  9049  marypha1  9424  m2cpmf1  22226  ausgrusgri  28407  uspgrupgrushgr  28416  usgrumgruspgr  28419  usgruspgrb  28420  usgrres  28544  usgrres1  28551  lindflbs  32459  dimkerim  32656  cantnfub2  42004
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