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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 16366 | . . 3 ⊢ ((𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) | |
4 | 1, 2, 3 | syl2anc 576 | . 2 ⊢ (𝜑 → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
5 | wunsets.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1 | wunres 9949 | . . 3 ⊢ (𝜑 → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ 𝑈) |
7 | 5, 2 | wunsn 9934 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
8 | 5, 6, 7 | wunun 9928 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ 𝑈) |
9 | 4, 8 | eqeltrd 2859 | 1 ⊢ (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 Vcvv 3408 ∖ cdif 3819 ∪ cun 3820 {csn 4435 dom cdm 5403 ↾ cres 5405 (class class class)co 6974 WUnicwun 9918 sSet csts 16335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-tr 5027 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-res 5415 df-iota 6149 df-fun 6187 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-wun 9920 df-sets 16344 |
This theorem is referenced by: wunress 16418 catcoppccl 17238 |
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