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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 9843 | . . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
4 | 1, 3 | wununi 9843 | . 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈) |
5 | ssun1 4003 | . . . 4 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
6 | dmrnssfld 5617 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
7 | 5, 6 | sstri 3836 | . . 3 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴 |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → dom 𝐴 ⊆ ∪ ∪ 𝐴) |
9 | 1, 4, 8 | wunss 9849 | 1 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 ∪ cun 3796 ⊆ wss 3798 ∪ cuni 4658 dom cdm 5342 ran crn 5343 WUnicwun 9837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-tr 4976 df-cnv 5350 df-dm 5352 df-rn 5353 df-wun 9839 |
This theorem is referenced by: wuncnv 9867 wunco 9870 wuntpos 9871 catcoppccl 17110 |
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