Theorem opabssxp 4838

Description: An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)

Assertion

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

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

Allowed substitution hints:   φ(x,y)

Proof of Theorem opabssxp

Step	Hyp	Ref	Expression
1		simpl 443	. . 3 ⊢ (((x ∈ A ∧ y ∈ B) ∧ φ) → (x ∈ A ∧ y ∈ B))
2	1	ssopab2i 4715	. 2 ⊢ {⟨x, y⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)} ⊆ {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)}
3		df-xp 4785	. 2 ⊢ (A × B) = {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)}
4	2, 3	sseqtr4i 3305	1 ⊢ {⟨x, y⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)} ⊆ (A × B)

Syntax hints:   ∧ wa 358   ∈ wcel 1710   ⊆ wss 3258  {copab 4623   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-xp 4785

This theorem is referenced by:  dmoprabss  5576