Theorem dmoprabss 6102

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 6101	. 2 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}
2		19.42v 1955	. . . 4 ⊢ (∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑧𝜑))
3	2	opabbii 4156	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑧𝜑)}
4		opabssxp 4800	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑧𝜑)} ⊆ (𝐴 × 𝐵)
5	3, 4	eqsstri 3259	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
6	1, 5	eqsstri 3259	1 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)

Syntax hints:   ∧ wa 104  ∃wex 1540   ∈ wcel 2202   ⊆ wss 3200  {copab 4149   × cxp 4723  dom cdm 4725  {coprab 6018

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-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  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-br 4089  df-opab 4151  df-xp 4731  df-dm 4735  df-oprab 6021

This theorem is referenced by:  elmpocl  6216  oprabexd  6288  oprabex  6289  elmpom  6402  axaddf  8087  axmulf  8088