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Mirrors > Home > MPE Home > Th. List > wunsets | Structured version Visualization version GIF version |
Description: Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
wunsets.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunsets.2 | ⊢ (𝜑 → 𝑆 ∈ 𝑈) |
wunsets.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunsets | ⊢ (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wunsets.2 | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑈) | |
2 | wunsets.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | setsvalg 15719 | . . 3 ⊢ ((𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) | |
4 | 1, 2, 3 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
5 | wunsets.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1 | wunres 9432 | . . 3 ⊢ (𝜑 → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ 𝑈) |
7 | 5, 2 | wunsn 9417 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
8 | 5, 6, 7 | wunun 9411 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ 𝑈) |
9 | 4, 8 | eqeltrd 2688 | 1 ⊢ (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 {csn 4125 dom cdm 5038 ↾ cres 5040 (class class class)co 6549 WUnicwun 9401 sSet csts 15693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wun 9403 df-sets 15701 |
This theorem is referenced by: wunress 15767 catcoppccl 16581 |
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