Theorem resoprab 5741

Description: Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)

Assertion

Ref	Expression
resoprab	⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}

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

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

Proof of Theorem resoprab

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resopab 4756	. . 3 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↾ (𝐴 × 𝐵)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
2		19.42vv 1836	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3		an12 528	. . . . . . 7 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)))
4		eleq1 2150	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
5		opelxp 4467	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
6	4, 5	syl6bb 194	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
7	6	anbi1d 453	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
8	7	pm5.32i 442	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
9	3, 8	bitri 182	. . . . . 6 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
10	9	2exbii 1542	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
11	2, 10	bitr3i 184	. . . 4 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
12	11	opabbii 3905	. . 3 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))}
13	1, 12	eqtri 2108	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↾ (𝐴 × 𝐵)) = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))}
14		dfoprab2 5696	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15	14	reseq1i 4709	. 2 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = ({⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↾ (𝐴 × 𝐵))
16		dfoprab2 5696	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))}
17	13, 15, 16	3eqtr4i 2118	1 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}

Syntax hints:   ∧ wa 102   = wceq 1289  ∃wex 1426   ∈ wcel 1438  ⟨cop 3449  {copab 3898   × cxp 4436   ↾ cres 4440  {coprab 5653

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444  df-rel 4445  df-res 4450  df-oprab 5656

This theorem is referenced by:  resoprab2  5742