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Theorem opabssxpd 5736
Description: An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 8081. (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 5211 . 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 5728 . . . . . . 7 ((𝜑𝜓) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
65adantrl 716 . . . . . 6 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
72, 6eqeltrd 2839 . . . . 5 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → 𝑧 ∈ (𝐴 × 𝐵))
87ex 412 . . . 4 (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
98exlimdvv 1932 . . 3 (𝜑 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
109abssdv 4078 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} ⊆ (𝐴 × 𝐵))
111, 10eqsstrid 4044 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wss 3963  cop 4637  {copab 5210   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695
This theorem is referenced by:  opabex2  8081  erlval  33245  uspgropssxp  47988
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