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Theorem f1ss 5206
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 5200 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fss 5157 . . 3 ((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
31, 2sylan 277 . 2 ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
4 df-f1 5007 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
54simprbi 269 . . 3 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
65adantr 270 . 2 ((𝐹:𝐴1-1𝐵𝐵𝐶) → Fun 𝐹)
7 df-f1 5007 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
83, 6, 7sylanbrc 408 1 ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴1-1𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wss 2997  ccnv 4427  Fun wfun 4996  wf 4998  1-1wf1 4999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-f 5006  df-f1 5007
This theorem is referenced by: (None)
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