Theorem resoprab2 7529

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 7528	. 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 469	. . . 4 ⊢ ((((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
3		an4 654	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)))
4		ssel 3975	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
5	4	pm4.71d 562	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴)))
6	5	bicomd 222	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐶))
7		ssel 3975	. . . . . . . . 9 ⊢ (𝐷 ⊆ 𝐵 → (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐵))
8	7	pm4.71d 562	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐵 → (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)))
9	8	bicomd 222	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐵 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵) ↔ 𝑦 ∈ 𝐷))
10	6, 9	bi2anan9 637	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
11	3, 10	bitrid 282	. . . . 5 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
12	11	anbi1d 630	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)))
13	2, 12	bitr3id 284	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)))
14	13	oprabbidv 7477	. 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 2784	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 396   = wceq 1541   ∈ wcel 2106   ⊆ wss 3948   × cxp 5674   ↾ cres 5678  {coprab 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-xp 5682  df-rel 5683  df-res 5688  df-oprab 7415

This theorem is referenced by:  resmpo  7530