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Mirrors > Home > MPE Home > Th. List > f1ssr | Structured version Visualization version GIF version |
Description: A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
Ref | Expression |
---|---|
f1ssr | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1fn 6102 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹 Fn 𝐴) |
3 | simpr 477 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐶) | |
4 | df-f 5892 | . . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
5 | 2, 3, 4 | sylanbrc 698 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) |
6 | df-f1 5893 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
7 | 6 | simprbi 480 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹) |
8 | 7 | adantr 481 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → Fun ◡𝐹) |
9 | df-f1 5893 | . 2 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹)) | |
10 | 5, 8, 9 | sylanbrc 698 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ⊆ wss 3574 ◡ccnv 5113 ran crn 5115 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 –1-1→wf1 5885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 197 df-an 386 df-f 5892 df-f1 5893 |
This theorem is referenced by: domdifsn 8043 marypha1 8340 m2cpmf1 20548 ausgrusgri 26063 uspgrupgrushgr 26072 usgrumgruspgr 26075 usgruspgrb 26076 usgrres 26200 usgrres1 26207 |
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