Theorem resoprab2 6032

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 6031	. 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 401	. . . 4 ⊢ ((((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
3		an4 586	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)))
4		ssel 3186	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
5	4	pm4.71d 393	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴)))
6	5	bicomd 141	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐶))
7		ssel 3186	. . . . . . . . 9 ⊢ (𝐷 ⊆ 𝐵 → (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐵))
8	7	pm4.71d 393	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐵 → (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)))
9	8	bicomd 141	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐵 → ((𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵) ↔ 𝑦 ∈ 𝐷))
10	6, 9	bi2anan9 606	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
11	3, 10	bitrid 192	. . . . 5 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
12	11	anbi1d 465	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)))
13	2, 12	bitr3id 194	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)))
14	13	oprabbidv 5989	. 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 2249	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 104   = wceq 1372   ∈ wcel 2175   ⊆ wss 3165   × cxp 4671   ↾ cres 4675  {coprab 5935

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-opab 4105  df-xp 4679  df-rel 4680  df-res 4685  df-oprab 5938

This theorem is referenced by:  resmpo  6033