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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 6018 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
| 2 | 1 | rnssi 5950 | 1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊆ wss 3950 ran crn 5685 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 | 
| This theorem is referenced by: imadifssran 6170 gsumhashmul 33065 nelrnres 45197 limsupvaluz2 45758 supcnvlimsup 45760 limsupgtlem 45797 sge0split 46429 | 
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