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Theorem opabssxpd 5716
Description: An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 8042. (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 5204 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
2 simprl 768 . . . . . 6 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → 𝑧 = ⟨𝑥, 𝑦⟩)
3 opabssxpd.x . . . . . . . 8 ((𝜑𝜓) → 𝑥𝐴)
4 opabssxpd.y . . . . . . . 8 ((𝜑𝜓) → 𝑦𝐵)
53, 4opelxpd 5708 . . . . . . 7 ((𝜑𝜓) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
65adantrl 713 . . . . . 6 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
72, 6eqeltrd 2827 . . . . 5 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → 𝑧 ∈ (𝐴 × 𝐵))
87ex 412 . . . 4 (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
98exlimdvv 1929 . . 3 (𝜑 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
109abssdv 4060 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} ⊆ (𝐴 × 𝐵))
111, 10eqsstrid 4025 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wss 3943  cop 4629  {copab 5203   × cxp 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-xp 5675
This theorem is referenced by:  opabex2  8042  uspgropssxp  47094
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