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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 10683 | . . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
4 | 1, 3 | wununi 10683 | . 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈) |
5 | ssun2 4169 | . . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
6 | dmrnssfld 5961 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
7 | 5, 6 | sstri 3987 | . . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ran 𝐴 ⊆ ∪ ∪ 𝐴) |
9 | 1, 4, 8 | wunss 10689 | 1 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∪ cun 3942 ⊆ wss 3944 ∪ cuni 4901 dom cdm 5669 ran crn 5670 WUnicwun 10677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 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-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-tr 5259 df-cnv 5677 df-dm 5679 df-rn 5680 df-wun 10679 |
This theorem is referenced by: wuncnv 10707 wunfv 10709 wunco 10710 wuntpos 10711 wunstr 17103 wunfunc 17831 wunfuncOLD 17832 wunnat 17889 wunnatOLD 17890 catcoppccl 18049 catcoppcclOLD 18050 catcfuccl 18051 catcfucclOLD 18052 catcxpccl 18141 catcxpcclOLD 18142 |
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