Theorem dfoprab4 8080

Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfoprab4.1	⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dfoprab4	⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}

Distinct variable groups:   𝑥,𝑤,𝑦,𝐴   𝑤,𝐵,𝑥,𝑦   𝜑,𝑥,𝑦   𝜓,𝑤   𝑧,𝑤,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧)   𝐴(𝑧)   𝐵(𝑧)

Proof of Theorem dfoprab4

Step	Hyp	Ref	Expression
1		xpss 5701	. . . . . 6 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
2	1	sseli 3979	. . . . 5 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 ∈ (V × V))
3	2	adantr 480	. . . 4 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) → 𝑤 ∈ (V × V))
4	3	pm4.71ri 560	. . 3 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)))
5	4	opabbii 5210	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))}
6		eleq1 2829	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
7		opelxp 5721	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
8	6, 7	bitrdi 287	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
9		dfoprab4.1	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
10	8, 9	anbi12d 632	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)))
11	10	dfoprab3 8079	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
12	5, 11	eqtri 2765	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ⟨cop 4632  {copab 5205   × cxp 5683  {coprab 7432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-oprab 7435  df-1st 8014  df-2nd 8015

This theorem is referenced by:  dfoprab4f  8081  dfxp3  8086  xrninxp  38393