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Mirrors > Home > MPE Home > Th. List > f1resf1 | Structured version Visualization version GIF version |
Description: The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
f1resf1 | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ssres 6806 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) | |
2 | 1 | 3adant3 1129 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) |
3 | frn 6734 | . . 3 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐷 → ran (𝐹 ↾ 𝐶) ⊆ 𝐷) | |
4 | 3 | 3ad2ant3 1132 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → ran (𝐹 ↾ 𝐶) ⊆ 𝐷) |
5 | f1ssr 6805 | . 2 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ ran (𝐹 ↾ 𝐶) ⊆ 𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷) | |
6 | 2, 4, 5 | syl2anc 582 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ⊆ wss 3949 ran crn 5683 ↾ cres 5684 ⟶wf 6549 –1-1→wf1 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 |
This theorem is referenced by: inlresf1 9946 inrresf1 9948 pfxf1 32686 aks6d1c2 41633 |
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