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Mirrors > Home > MPE Home > Th. List > wuncval | Structured version Visualization version GIF version |
Description: Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wuncval | ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | wunex 9440 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑢 ∈ WUni 𝐴 ⊆ 𝑢) | |
3 | rabn0 3912 | . . . 4 ⊢ ({𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ≠ ∅ ↔ ∃𝑢 ∈ WUni 𝐴 ⊆ 𝑢) | |
4 | 2, 3 | sylibr 223 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ≠ ∅) |
5 | intex 4747 | . . 3 ⊢ ({𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ≠ ∅ ↔ ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ∈ V) | |
6 | 4, 5 | sylib 207 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ∈ V) |
7 | sseq1 3589 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑢 ↔ 𝐴 ⊆ 𝑢)) | |
8 | 7 | rabbidv 3164 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑢 ∈ WUni ∣ 𝑥 ⊆ 𝑢} = {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢}) |
9 | 8 | inteqd 4415 | . . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑢 ∈ WUni ∣ 𝑥 ⊆ 𝑢} = ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢}) |
10 | df-wunc 9404 | . . 3 ⊢ wUniCl = (𝑥 ∈ V ↦ ∩ {𝑢 ∈ WUni ∣ 𝑥 ⊆ 𝑢}) | |
11 | 9, 10 | fvmptg 6189 | . 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢} ∈ V) → (wUniCl‘𝐴) = ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢}) |
12 | 1, 6, 11 | syl2anc 691 | 1 ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ∩ cint 4410 ‘cfv 5804 WUnicwun 9401 wUniClcwunm 9402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-wun 9403 df-wunc 9404 |
This theorem is referenced by: wuncid 9444 wunccl 9445 wuncss 9446 |
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