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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 6006 | . . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ⊆ 𝐴) |
5 | 1, 2, 4 | wunss 10713 | 1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3948 ↾ cres 5678 WUnicwun 10701 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1088 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-in 3955 df-ss 3965 df-pw 4604 df-uni 4909 df-tr 5266 df-res 5688 df-wun 10703 |
This theorem is referenced by: wunsets 17117 |
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