Theorem resoprab2 7552

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 7551	. 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 468	. . . 4 ⊢ ((((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
3		an4 656	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)))
4		ssel 3989	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
5	4	pm4.71d 561	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴)))
6	5	bicomd 223	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐶))
7		ssel 3989	. . . . . . . . 9 ⊢ (𝐷 ⊆ 𝐵 → (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐵))
8	7	pm4.71d 561	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐵 → (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)))
9	8	bicomd 223	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐵 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵) ↔ 𝑦 ∈ 𝐷))
10	6, 9	bi2anan9 638	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
11	3, 10	bitrid 283	. . . . 5 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
12	11	anbi1d 631	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)))
13	2, 12	bitr3id 285	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)))
14	13	oprabbidv 7499	. 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	eqtrid 2787	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 395   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963   × cxp 5687   ↾ cres 5691  {coprab 7432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696  df-res 5701  df-oprab 7435

This theorem is referenced by:  resmpo  7553