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Theorem opabssxpd 5757
Description: An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 7741. (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,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,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 5096 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
2 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → 𝑧 = ⟨𝑥, 𝑦⟩)
3 opabssxpd.x . . . . . . . 8 ((𝜑𝜓) → 𝑥𝐴)
4 opabssxpd.y . . . . . . . 8 ((𝜑𝜓) → 𝑦𝐵)
53, 4opelxpd 5561 . . . . . . 7 ((𝜑𝜓) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
65adantrl 715 . . . . . 6 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
72, 6eqeltrd 2893 . . . . 5 ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → 𝑧 ∈ (𝐴 × 𝐵))
87ex 416 . . . 4 (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
98exlimdvv 1935 . . 3 (𝜑 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
109abssdv 3999 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} ⊆ (𝐴 × 𝐵))
111, 10eqsstrid 3966 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wss 3884  cop 4534  {copab 5095   × cxp 5521
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-pr 5298
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-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-xp 5529
This theorem is referenced by:  opabex2  7741  uspgropssxp  44359
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