Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6015 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹 Fn 𝐴) |
3 | simpr 476 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ran 𝐹 ⊆ 𝐶) | |
4 | df-f 5808 | . . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
5 | 2, 3, 4 | sylanbrc 695 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) |
6 | df-f1 5809 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
7 | 6 | simprbi 479 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹) |
8 | 7 | adantr 480 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → Fun ◡𝐹) |
9 | df-f1 5809 | . 2 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹)) | |
10 | 5, 8, 9 | sylanbrc 695 | 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 383 ⊆ wss 3540 ◡ccnv 5037 ran crn 5039 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 |
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 196 df-an 385 df-f 5808 df-f1 5809 |
This theorem is referenced by: domdifsn 7928 marypha1 8223 m2cpmf1 20367 usgrares1 25939 ausgrusgri 40398 uspgrupgrushgr 40407 usgrumgruspgr 40410 usgruspgrb 40411 usgrres1 40534 |
Copyright terms: Public domain | W3C validator |