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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwiuncl | Structured version Visualization version GIF version |
Description: Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.) |
Ref | Expression |
---|---|
elpwiuncl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elpwiuncl.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝒫 𝐶) |
Ref | Expression |
---|---|
elpwiuncl | ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwiuncl.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝒫 𝐶) | |
2 | 1 | elpwid 4118 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
3 | 2 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶) |
4 | iunss 4497 | . . 3 ⊢ (∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶) | |
5 | 3, 4 | sylibr 223 | . 2 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶) |
6 | elpwiuncl.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | 1 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶) |
8 | 6, 7 | jca 553 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶)) |
9 | iunexg 7035 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ V) | |
10 | elpwg 4116 | . . 3 ⊢ (∪ 𝑘 ∈ 𝐴 𝐵 ∈ V → (∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶 ↔ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶)) | |
11 | 8, 9, 10 | 3syl 18 | . 2 ⊢ (𝜑 → (∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶 ↔ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶)) |
12 | 5, 11 | mpbird 246 | 1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∪ ciun 4455 |
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-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: carsggect 29707 carsgclctunlem2 29708 |
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