Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1ss GIF version

Theorem f1ss 5097
 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 5092 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fss 5054 . . 3 ((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
31, 2sylan 267 . 2 ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
4 df-f1 4907 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
54simprbi 260 . . 3 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
65adantr 261 . 2 ((𝐹:𝐴1-1𝐵𝐵𝐶) → Fun 𝐹)
7 df-f1 4907 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
83, 6, 7sylanbrc 394 1 ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴1-1𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ⊆ wss 2917  ◡ccnv 4344  Fun wfun 4896  ⟶wf 4898  –1-1→wf1 4899 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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906  df-f1 4907 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator