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| Description: A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| f1ssres | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | f1f 6803 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
| 2 | fssres 6773 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) | |
| 3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) | 
| 4 | df-f1 6565 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
| 5 | funres11 6642 | . . . 4 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐶)) | |
| 6 | 4, 5 | simplbiim 504 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡(𝐹 ↾ 𝐶)) | 
| 7 | 6 | adantr 480 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡(𝐹 ↾ 𝐶)) | 
| 8 | df-f1 6565 | . 2 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶))) | |
| 9 | 3, 7, 8 | sylanbrc 583 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3950 ◡ccnv 5683 ↾ cres 5686 Fun wfun 6554 ⟶wf 6556 –1-1→wf1 6557 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 | 
| This theorem is referenced by: f1resf1 6811 f1ores 6861 oacomf1olem 8603 domssl 9039 undom 9100 pwfseqlem5 10704 hashimarn 14480 hashf1lem2 14496 conjsubgen 19270 sylow1lem2 19618 sylow2blem1 19639 usgrres 29326 lmimdim 33655 | 
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