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Theorem f1ss 5040
 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 B𝐶) → 𝐹:A1-1𝐶)

Proof of Theorem f1ss
StepHypRef Expression
1 f1f 5035 . . 3 (𝐹:A1-1B𝐹:AB)
2 fss 4997 . . 3 ((𝐹:AB B𝐶) → 𝐹:A𝐶)
31, 2sylan 267 . 2 ((𝐹:A1-1B B𝐶) → 𝐹:A𝐶)
4 df-f1 4850 . . . 4 (𝐹:A1-1B ↔ (𝐹:AB Fun 𝐹))
54simprbi 260 . . 3 (𝐹:A1-1B → Fun 𝐹)
65adantr 261 . 2 ((𝐹:A1-1B B𝐶) → Fun 𝐹)
7 df-f1 4850 . 2 (𝐹:A1-1𝐶 ↔ (𝐹:A𝐶 Fun 𝐹))
83, 6, 7sylanbrc 394 1 ((𝐹:A1-1B B𝐶) → 𝐹:A1-1𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ⊆ wss 2911  ◡ccnv 4287  Fun wfun 4839  ⟶wf 4841  –1-1→wf1 4842 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849  df-f1 4850 This theorem is referenced by: (None)
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