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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 17204 | . . 3 ⊢ ((𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) | |
| 4 | 1, 2, 3 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) | 
| 5 | wunsets.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
| 6 | 5, 1 | wunres 10772 | . . 3 ⊢ (𝜑 → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ 𝑈) | 
| 7 | 5, 2 | wunsn 10757 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) | 
| 8 | 5, 6, 7 | wunun 10751 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ 𝑈) | 
| 9 | 4, 8 | eqeltrd 2840 | 1 ⊢ (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∖ cdif 3947 ∪ cun 3948 {csn 4625 dom cdm 5684 ↾ cres 5686 (class class class)co 7432 WUnicwun 10741 sSet csts 17201 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-wun 10743 df-sets 17202 | 
| This theorem is referenced by: wunress 17296 wunressOLD 17297 catcoppccl 18163 | 
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