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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 16694 | . . 3 ⊢ ((𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) | |
4 | 1, 2, 3 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
5 | wunsets.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1 | wunres 10310 | . . 3 ⊢ (𝜑 → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ 𝑈) |
7 | 5, 2 | wunsn 10295 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
8 | 5, 6, 7 | wunun 10289 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ 𝑈) |
9 | 4, 8 | eqeltrd 2831 | 1 ⊢ (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∖ cdif 3850 ∪ cun 3851 {csn 4527 dom cdm 5536 ↾ cres 5538 (class class class)co 7191 WUnicwun 10279 sSet csts 16664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-res 5548 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-wun 10281 df-sets 16673 |
This theorem is referenced by: wunress 16748 catcoppccl 17577 catcoppcclOLD 17578 |
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