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Mirrors > Home > MPE Home > Th. List > f1ssres | Structured version Visualization version GIF version |
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 6787 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | fssres 6757 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) | |
3 | 1, 2 | sylan 579 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
4 | df-f1 6548 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
5 | funres11 6625 | . . . 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 6548 | . 2 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶))) | |
9 | 3, 7, 8 | sylanbrc 582 | 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 3948 ◡ccnv 5675 ↾ cres 5678 Fun wfun 6537 ⟶wf 6539 –1-1→wf1 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 |
This theorem is referenced by: f1resf1 6796 f1ores 6847 oacomf1olem 8567 domssl 8997 undom 9062 pwfseqlem5 10661 hashimarn 14405 hashf1lem2 14422 conjsubgen 19166 sylow1lem2 19509 sylow2blem1 19530 usgrres 28833 lmimdim 32977 |
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