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Theorem f1ss 5262
 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:A1-1B B C) → F:A1-1C)

Proof of Theorem f1ss
StepHypRef Expression
1 f1f 5258 . . 3 (F:A1-1BF:A–→B)
2 fss 5230 . . 3 ((F:A–→B B C) → F:A–→C)
31, 2sylan 457 . 2 ((F:A1-1B B C) → F:A–→C)
4 df-f1 4792 . . . 4 (F:A1-1B ↔ (F:A–→B Fun F))
54simprbi 450 . . 3 (F:A1-1B → Fun F)
65adantr 451 . 2 ((F:A1-1B B C) → Fun F)
7 df-f1 4792 . 2 (F:A1-1C ↔ (F:A–→C Fun F))
83, 6, 7sylanbrc 645 1 ((F:A1-1B B C) → F:A1-1C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ⊆ wss 3257  ◡ccnv 4771  Fun wfun 4775  –→wf 4777  –1-1→wf1 4778 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791  df-f1 4792 This theorem is referenced by:  dflec3  6221
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