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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-imdirval2lem | Structured version Visualization version GIF version |
Description: Lemma for bj-imdirval2 34863 and bj-iminvval2 34874. (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 5241 | . . 3 ⊢ (𝜑 → 𝒫 𝐴 ∈ V) |
3 | bj-imdirval2lem.exb | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | 3 | pwexd 5241 | . . 3 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
5 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ⊆ 𝐴) | |
6 | velpw 4492 | . . . 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 4492 | . . . 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 7752 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)} ∈ V) |
12 | simpl 487 | . . . 4 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓) → (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) | |
13 | 12 | ssopab2i 5400 | . . 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 5187 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∈ wcel 2112 Vcvv 3407 ⊆ wss 3854 𝒫 cpw 4487 {copab 5087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-opab 5088 df-xp 5523 df-rel 5524 |
This theorem is referenced by: bj-imdirval2 34863 bj-iminvval2 34874 |
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