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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnresss | Structured version Visualization version GIF version |
Description: The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
rnresss | ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resss 5671 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
2 | rnss 5599 | . 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3792 ran crn 5356 ↾ cres 5357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-cnv 5363 df-dm 5365 df-rn 5366 df-res 5367 |
This theorem is referenced by: nelrnres 40301 limsupvaluz2 40882 supcnvlimsup 40884 limsupgtlem 40921 sge0split 41554 |
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