Theorem resoprab 7568

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 6063	. . 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 1957	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3		an12 644	. . . . . . 7 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)))
4		eleq1 2832	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
5		opelxp 5736	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
6	4, 5	bitrdi 287	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
7	6	anbi1d 630	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
8	7	pm5.32i 574	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
9	3, 8	bitri 275	. . . . . 6 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
10	9	2exbii 1847	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
11	2, 10	bitr3i 277	. . . 4 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)))
12	11	opabbii 5233	. . 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 2768	. 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 7508	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15	14	reseq1i 6005	. 2 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = ({⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↾ (𝐴 × 𝐵))
16		dfoprab2 7508	. 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 2778	1 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ⟨cop 4654  {copab 5228   × cxp 5698   ↾ cres 5702  {coprab 7449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-rel 5707  df-res 5712  df-oprab 7452

This theorem is referenced by:  resoprab2  7569  df1stres  32715  df2ndres  32716