Theorem bj-imdirval2lem 37713

Description: Lemma for bj-imdirval2 37714 and bj-iminvval2 37725. (Contributed by BJ, 23-May-2024.)

Hypotheses

Ref	Expression
bj-imdirval2lem.exa	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-imdirval2lem.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
bj-imdirval2lem	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem bj-imdirval2lem

Step	Hyp	Ref	Expression
1		bj-imdirval2lem.exa	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2	1	pwexd 5351	. . 3 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
3		bj-imdirval2lem.exb	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
4	3	pwexd 5351	. . 3 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
5		simprl 782	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ⊆ 𝐴)
6		velpw 4572	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7	5, 6	sylibr 237	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ∈ 𝒫 𝐴)
8		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ⊆ 𝐵)
9		velpw 4572	. . . 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 8053	. 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 5536	. . 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 5295	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567  {copab 5177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-opab 5178  df-xp 5668  df-rel 5669

This theorem is referenced by:  bj-imdirval2  37714  bj-iminvval2  37725