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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 5876 | . . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ⊆ 𝐴) |
5 | 1, 2, 4 | wunss 10326 | 1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3866 ↾ cres 5553 WUnicwun 10314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 ax-sep 5192 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rab 3070 df-v 3410 df-in 3873 df-ss 3883 df-pw 4515 df-uni 4820 df-tr 5162 df-res 5563 df-wun 10316 |
This theorem is referenced by: wunsets 16730 |
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