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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 10746 | . . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
| 4 | 1, 3 | wununi 10746 | . 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈) |
| 5 | ssun2 4179 | . . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
| 6 | dmrnssfld 5984 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
| 7 | 5, 6 | sstri 3993 | . . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
| 8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ran 𝐴 ⊆ ∪ ∪ 𝐴) |
| 9 | 1, 4, 8 | wunss 10752 | 1 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 ∪ cuni 4907 dom cdm 5685 ran crn 5686 WUnicwun 10740 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-cnv 5693 df-dm 5695 df-rn 5696 df-wun 10742 |
| This theorem is referenced by: wuncnv 10770 wunfv 10772 wunco 10773 wuntpos 10774 wunstr 17225 wunfunc 17946 wunnat 18004 catcoppccl 18162 catcfuccl 18163 catcxpccl 18252 |
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