Theorem dmoprabss 7537

Description: The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)

Assertion

Ref	Expression
dmoprabss	⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem dmoprabss

Step	Hyp	Ref	Expression
1		dmoprab 7536	. 2 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}
2		19.42v 1953	. . . 4 ⊢ (∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑧𝜑))
3	2	opabbii 5210	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑧𝜑)}
4		opabssxp 5778	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑧𝜑)} ⊆ (𝐴 × 𝐵)
5	3, 4	eqsstri 4030	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
6	1, 5	eqsstri 4030	1 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)

Syntax hints:   ∧ wa 395  ∃wex 1779   ∈ wcel 2108   ⊆ wss 3951  {copab 5205   × cxp 5683  dom cdm 5685  {coprab 7432

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-10 2141  ax-11 2157  ax-12 2177  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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  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-br 5144  df-opab 5206  df-xp 5691  df-dm 5695  df-oprab 7435

This theorem is referenced by:  mpondm0  7673  elmpocl  7674  oprabexd  8000  oprabex  8001  bropopvvv  8115  bropfvvvv  8117  dmaddsr  11125  dmmulsr  11126  axaddf  11185  axmulf  11186