Theorem resoprab 6116

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 5057	. . 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 1960	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3		an12 563	. . . . . . 7 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)))
4		eleq1 2294	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
5		opelxp 4755	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
6	4, 5	bitrdi 196	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
7	6	anbi1d 465	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
8	7	pm5.32i 454	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
9	3, 8	bitri 184	. . . . . 6 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
10	9	2exbii 1654	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
11	2, 10	bitr3i 186	. . . 4 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
12	11	opabbii 4156	. . 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 2252	. 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 6067	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15	14	reseq1i 5009	. 2 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = ({⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↾ (𝐴 × 𝐵))
16		dfoprab2 6067	. 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 2262	1 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}

Syntax hints:   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ⟨cop 3672  {copab 4149   × cxp 4723   ↾ cres 4727  {coprab 6018

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731  df-rel 4732  df-res 4737  df-oprab 6021

This theorem is referenced by:  resoprab2  6117