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 5861 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
2 | imass1 5949 | . 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 3853 ↾ cres 5538 “ cima 5539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 |
This theorem is referenced by: limsupres 42864 limsupresxr 42925 liminfresxr 42926 |
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