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Mirrors > Home > MPE Home > Th. List > wundm | Structured version Visualization version GIF version |
Description: A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wundm | ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | 1, 2 | wununi 10731 | . . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
4 | 1, 3 | wununi 10731 | . 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈) |
5 | ssun1 4170 | . . . 4 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
6 | dmrnssfld 5973 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
7 | 5, 6 | sstri 3986 | . . 3 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴 |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → dom 𝐴 ⊆ ∪ ∪ 𝐴) |
9 | 1, 4, 8 | wunss 10737 | 1 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∪ cun 3942 ⊆ wss 3944 ∪ cuni 4909 dom cdm 5678 ran crn 5679 WUnicwun 10725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-cnv 5686 df-dm 5688 df-rn 5689 df-wun 10727 |
This theorem is referenced by: wuncnv 10755 wunco 10758 wuntpos 10759 catcoppccl 18109 catcoppcclOLD 18110 |
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