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Theorem opabssxpd 5732
Description: An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 8082. (Contributed by AV, 26-Nov-2021.)
Hypotheses
Ref Expression
opabssxpd.x ((𝜑𝜓) → 𝑥𝐴)
opabssxpd.y ((𝜑𝜓) → 𝑦𝐵)
Assertion
Ref Expression
opabssxpd (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑥,𝐵   𝑦,𝐵   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem opabssxpd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-opab 5206 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
2 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → 𝑧 = ⟨𝑥, 𝑦⟩)
3 opabssxpd.x . . . . . . . 8 ((𝜑𝜓) → 𝑥𝐴)
4 opabssxpd.y . . . . . . . 8 ((𝜑𝜓) → 𝑦𝐵)
53, 4opelxpd 5724 . . . . . . 7 ((𝜑𝜓) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
65adantrl 716 . . . . . 6 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
72, 6eqeltrd 2841 . . . . 5 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → 𝑧 ∈ (𝐴 × 𝐵))
87ex 412 . . . 4 (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
98exlimdvv 1934 . . 3 (𝜑 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
109abssdv 4068 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} ⊆ (𝐴 × 𝐵))
111, 10eqsstrid 4022 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wss 3951  cop 4632  {copab 5205   × cxp 5683
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-pr 5432
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-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-xp 5691
This theorem is referenced by:  opabex2  8082  erlval  33262  uspgropssxp  48060
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