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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-imdirval2lem | Structured version Visualization version GIF version |
Description: Lemma for bj-imdirval2 35046 and bj-iminvval2 35057. (Contributed by BJ, 23-May-2024.) |
Ref | Expression |
---|---|
bj-imdirval2lem.exa | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
bj-imdirval2lem.exb | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
bj-imdirval2lem | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-imdirval2lem.exa | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
2 | 1 | pwexd 5261 | . . 3 ⊢ (𝜑 → 𝒫 𝐴 ∈ V) |
3 | bj-imdirval2lem.exb | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | 3 | pwexd 5261 | . . 3 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
5 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ⊆ 𝐴) | |
6 | velpw 4508 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
7 | 5, 6 | sylibr 237 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ∈ 𝒫 𝐴) |
8 | simprr 773 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ⊆ 𝐵) | |
9 | velpw 4508 | . . . 4 ⊢ (𝑦 ∈ 𝒫 𝐵 ↔ 𝑦 ⊆ 𝐵) | |
10 | 8, 9 | sylibr 237 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ∈ 𝒫 𝐵) |
11 | 2, 4, 7, 10 | opabex2 7816 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)} ∈ V) |
12 | simpl 486 | . . . 4 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓) → (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) | |
13 | 12 | ssopab2i 5420 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)} |
14 | 13 | a1i 11 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)}) |
15 | 11, 14 | ssexd 5206 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 Vcvv 3401 ⊆ wss 3857 𝒫 cpw 4503 {copab 5105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-opab 5106 df-xp 5546 df-rel 5547 |
This theorem is referenced by: bj-imdirval2 35046 bj-iminvval2 35057 |
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