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Theorem f1ss 5124
Description: A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)
Assertion
Ref Expression
f1ss ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴1-1𝐶)

Proof of Theorem f1ss
StepHypRef Expression
1 f1f 5119 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fss 5081 . . 3 ((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
31, 2sylan 271 . 2 ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
4 df-f1 4934 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
54simprbi 264 . . 3 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
65adantr 265 . 2 ((𝐹:𝐴1-1𝐵𝐵𝐶) → Fun 𝐹)
7 df-f1 4934 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
83, 6, 7sylanbrc 402 1 ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴1-1𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wss 2944  ccnv 4371  Fun wfun 4923  wf 4925  1-1wf1 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2951  df-ss 2958  df-f 4933  df-f1 4934
This theorem is referenced by: (None)
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