Theorem dfoprab4 7997

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 5638	. . . . . 6 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
2	1	sseli 3927	. . . . 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 5163	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))}
6		eleq1 2822	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
7		opelxp 5658	. . . . 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 7996	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
12	5, 11	eqtri 2757	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ⟨cop 4584  {copab 5158   × cxp 5620  {coprab 7357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fv 6498  df-oprab 7360  df-1st 7931  df-2nd 7932

This theorem is referenced by:  dfoprab4f  7998  dfxp3  8003  xrninxp  38539