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Mirrors > Home > MPE Home > Th. List > wunrn | Structured version Visualization version GIF version |
Description: A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunrn | ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | 1, 2 | wununi 10743 | . . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
4 | 1, 3 | wununi 10743 | . 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈) |
5 | ssun2 4188 | . . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
6 | dmrnssfld 5986 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
7 | 5, 6 | sstri 4004 | . . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ran 𝐴 ⊆ ∪ ∪ 𝐴) |
9 | 1, 4, 8 | wunss 10749 | 1 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∪ cun 3960 ⊆ wss 3962 ∪ cuni 4911 dom cdm 5688 ran crn 5689 WUnicwun 10737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-cnv 5696 df-dm 5698 df-rn 5699 df-wun 10739 |
This theorem is referenced by: wuncnv 10767 wunfv 10769 wunco 10770 wuntpos 10771 wunstr 17221 wunfunc 17951 wunfuncOLD 17952 wunnat 18010 wunnatOLD 18011 catcoppccl 18170 catcoppcclOLD 18171 catcfuccl 18172 catcfucclOLD 18173 catcxpccl 18262 catcxpcclOLD 18263 |
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