Theorem resoprab2 5800

Description: Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
resoprab2	⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)})

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

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

Proof of Theorem resoprab2

Step	Hyp	Ref	Expression
1		resoprab 5799	. 2 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))}
2		anass 396	. . . 4 ⊢ ((((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
3		an4 556	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)))
4		ssel 3041	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
5	4	pm4.71d 388	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴)))
6	5	bicomd 140	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐶))
7		ssel 3041	. . . . . . . . 9 ⊢ (𝐷 ⊆ 𝐵 → (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐵))
8	7	pm4.71d 388	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐵 → (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)))
9	8	bicomd 140	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐵 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵) ↔ 𝑦 ∈ 𝐷))
10	6, 9	bi2anan9 576	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
11	3, 10	syl5bb 191	. . . . 5 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
12	11	anbi1d 456	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)))
13	2, 12	syl5bbr 193	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)))
14	13	oprabbidv 5757	. 2 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)})
15	1, 14	syl5eq 2144	1 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)})

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1299   ∈ wcel 1448   ⊆ wss 3021   × cxp 4475   ↾ cres 4479  {coprab 5707

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-opab 3930  df-xp 4483  df-rel 4484  df-res 4489  df-oprab 5710

This theorem is referenced by:  resmpo  5801