| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > resimass | Structured version Visualization version GIF version | ||
| Description: The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| resimass | ⊢ ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | resss 6019 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
| 2 | imass1 6119 | . 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ⊆ wss 3951 ↾ cres 5687 “ cima 5688 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: limsupres 45720 limsupresxr 45781 liminfresxr 45782 |
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