Theorem opabssxpd 4762

Description: An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 6357. (Contributed by AV, 26-Nov-2021.)

Hypotheses

Ref	Expression
opabssxpd.x	⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
opabssxpd.y	⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)

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.

Step	Hyp	Ref	Expression
1		df-opab 4151	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
2		simprl 531	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → 𝑧 = ⟨𝑥, 𝑦⟩)
3		opabssxpd.x	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
4		opabssxpd.y	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)
5	3, 4	opelxpd 4758	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
6	5	adantrl 478	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
7	2, 6	eqeltrd 2308	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → 𝑧 ∈ (𝐴 × 𝐵))
8	7	ex 115	. . . 4 ⊢ (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
9	8	exlimdvv 1946	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵)))
10	9	abssdv 3301	. 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} ⊆ (𝐴 × 𝐵))
11	1, 10	eqsstrid 3273	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {cab 2217   ⊆ wss 3200  ⟨cop 3672  {copab 4149   × cxp 4723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731

This theorem is referenced by:  opabex2  6357