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Theorem bj-imdirval2lem 34592
Description: Lemma for bj-imdirval2 34593 and bj-iminvval2 34604. (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 5248 . . 3 (𝜑 → 𝒫 𝐴 ∈ V)
3 bj-imdirval2lem.exb . . . 4 (𝜑𝐵𝑉)
43pwexd 5248 . . 3 (𝜑 → 𝒫 𝐵 ∈ V)
5 simprl 770 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑥𝐴)
6 velpw 4505 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
75, 6sylibr 237 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑥 ∈ 𝒫 𝐴)
8 simprr 772 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑦𝐵)
9 velpw 4505 . . . 4 (𝑦 ∈ 𝒫 𝐵𝑦𝐵)
108, 9sylibr 237 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑦 ∈ 𝒫 𝐵)
112, 4, 7, 10opabex2 7741 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∈ V)
12 simpl 486 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝜓) → (𝑥𝐴𝑦𝐵))
1312ssopab2i 5405 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
1413a1i 11 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
1511, 14ssexd 5195 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2112  Vcvv 3444  wss 3884  𝒫 cpw 4500  {copab 5095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-opab 5096  df-xp 5529  df-rel 5530
This theorem is referenced by:  bj-imdirval2  34593  bj-iminvval2  34604
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