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Mirrors > Home > MPE Home > Th. List > 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 6003 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
2 | 1 | rnssi 5938 | 1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3946 ran crn 5675 ↾ cres 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5146 df-opab 5208 df-cnv 5682 df-dm 5684 df-rn 5685 df-res 5686 |
This theorem is referenced by: gsumhashmul 32929 nelrnres 44830 limsupvaluz2 45395 supcnvlimsup 45397 limsupgtlem 45434 sge0split 46066 |
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