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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 16512 | . . 3 ⊢ ((𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) | |
4 | 1, 2, 3 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) |
5 | wunsets.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1 | wunres 10153 | . . 3 ⊢ (𝜑 → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ 𝑈) |
7 | 5, 2 | wunsn 10138 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈) |
8 | 5, 6, 7 | wunun 10132 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ 𝑈) |
9 | 4, 8 | eqeltrd 2913 | 1 ⊢ (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 ∪ cun 3934 {csn 4567 dom cdm 5555 ↾ cres 5557 (class class class)co 7156 WUnicwun 10122 sSet csts 16481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-wun 10124 df-sets 16490 |
This theorem is referenced by: wunress 16564 catcoppccl 17368 |
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