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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-imdirval2lem | Structured version Visualization version GIF version | ||
| Description: Lemma for bj-imdirval2 37184 and bj-iminvval2 37195. (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 5379 | . . 3 ⊢ (𝜑 → 𝒫 𝐴 ∈ V) |
| 3 | bj-imdirval2lem.exb | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
| 4 | 3 | pwexd 5379 | . . 3 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
| 5 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ⊆ 𝐴) | |
| 6 | velpw 4605 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
| 7 | 5, 6 | sylibr 234 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ∈ 𝒫 𝐴) |
| 8 | simprr 773 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ⊆ 𝐵) | |
| 9 | velpw 4605 | . . . 4 ⊢ (𝑦 ∈ 𝒫 𝐵 ↔ 𝑦 ⊆ 𝐵) | |
| 10 | 8, 9 | sylibr 234 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ∈ 𝒫 𝐵) |
| 11 | 2, 4, 7, 10 | opabex2 8082 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)} ∈ V) |
| 12 | simpl 482 | . . . 4 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓) → (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) | |
| 13 | 12 | ssopab2i 5555 | . . 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 5324 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 {copab 5205 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: bj-imdirval2 37184 bj-iminvval2 37195 |
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