Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > wunres | Structured version Visualization version GIF version |
Description: A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunres | ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | resss 5878 | . . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ⊆ 𝐴) |
5 | 1, 2, 4 | wunss 10134 | 1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⊆ wss 3936 ↾ cres 5557 WUnicwun 10122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-pw 4541 df-uni 4839 df-tr 5173 df-res 5567 df-wun 10124 |
This theorem is referenced by: wunsets 16524 |
Copyright terms: Public domain | W3C validator |