![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5843 | . 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
2 | 1 | rnssi 5774 | 1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3881 ran crn 5520 ↾ cres 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 |
This theorem is referenced by: gsumhashmul 30741 nelrnres 41814 limsupvaluz2 42380 supcnvlimsup 42382 limsupgtlem 42419 sge0split 43048 |
Copyright terms: Public domain | W3C validator |