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Theorem bj-imdirval2lem 37183
Description: Lemma for bj-imdirval2 37184 and bj-iminvval2 37195. (Contributed by BJ, 23-May-2024.)
Hypotheses
Ref Expression
bj-imdirval2lem.exa (𝜑𝐴𝑈)
bj-imdirval2lem.exb (𝜑𝐵𝑉)
Assertion
Ref Expression
bj-imdirval2lem (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 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
StepHypRef Expression
1 bj-imdirval2lem.exa . . . 4 (𝜑𝐴𝑈)
21pwexd 5379 . . 3 (𝜑 → 𝒫 𝐴 ∈ V)
3 bj-imdirval2lem.exb . . . 4 (𝜑𝐵𝑉)
43pwexd 5379 . . 3 (𝜑 → 𝒫 𝐵 ∈ V)
5 simprl 771 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑥𝐴)
6 velpw 4605 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
75, 6sylibr 234 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑥 ∈ 𝒫 𝐴)
8 simprr 773 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑦𝐵)
9 velpw 4605 . . . 4 (𝑦 ∈ 𝒫 𝐵𝑦𝐵)
108, 9sylibr 234 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑦 ∈ 𝒫 𝐵)
112, 4, 7, 10opabex2 8082 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∈ V)
12 simpl 482 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝜓) → (𝑥𝐴𝑦𝐵))
1312ssopab2i 5555 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
1413a1i 11 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
1511, 14ssexd 5324 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 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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