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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 17097 | . . 3 ⊢ ((𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) | |
4 | 1, 2, 3 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
5 | wunsets.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1 | wunres 10721 | . . 3 ⊢ (𝜑 → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ 𝑈) |
7 | 5, 2 | wunsn 10706 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
8 | 5, 6, 7 | wunun 10700 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ 𝑈) |
9 | 4, 8 | eqeltrd 2825 | 1 ⊢ (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∖ cdif 3937 ∪ cun 3938 {csn 4620 dom cdm 5666 ↾ cres 5668 (class class class)co 7401 WUnicwun 10690 sSet csts 17094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-tr 5256 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-res 5678 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-wun 10692 df-sets 17095 |
This theorem is referenced by: wunress 17193 wunressOLD 17194 catcoppccl 18068 catcoppcclOLD 18069 |
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