Theorem dfoprab4f 8080

Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 20-Dec-2008.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
dfoprab4f.x	⊢ Ⅎ𝑥𝜑
dfoprab4f.y	⊢ Ⅎ𝑦𝜑
dfoprab4f.1	⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dfoprab4f	⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}

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

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

Proof of Theorem dfoprab4f

Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1912	. . . . 5 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑡, 𝑢⟩
2		dfoprab4f.x	. . . . . 6 ⊢ Ⅎ𝑥𝜑
3		nfs1v 2154	. . . . . 6 ⊢ Ⅎ𝑥[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
4	2, 3	nfbi 1901	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
5	1, 4	nfim 1894	. . . 4 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
6		opeq1 4878	. . . . . 6 ⊢ (𝑥 = 𝑡 → ⟨𝑥, 𝑢⟩ = ⟨𝑡, 𝑢⟩)
7	6	eqeq2d 2746	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑤 = ⟨𝑥, 𝑢⟩ ↔ 𝑤 = ⟨𝑡, 𝑢⟩))
8		sbequ12 2249	. . . . . 6 ⊢ (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
9	8	bibi2d 342	. . . . 5 ⊢ (𝑥 = 𝑡 → ((𝜑 ↔ [𝑢 / 𝑦]𝜓) ↔ (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
10	7, 9	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑡 → ((𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓)) ↔ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))))
11		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑢⟩
12		dfoprab4f.y	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
13		nfs1v 2154	. . . . . . 7 ⊢ Ⅎ𝑦[𝑢 / 𝑦]𝜓
14	12, 13	nfbi 1901	. . . . . 6 ⊢ Ⅎ𝑦(𝜑 ↔ [𝑢 / 𝑦]𝜓)
15	11, 14	nfim 1894	. . . . 5 ⊢ Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
16		opeq2 4879	. . . . . . 7 ⊢ (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
17	16	eqeq2d 2746	. . . . . 6 ⊢ (𝑦 = 𝑢 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑢⟩))
18		sbequ12 2249	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
19	18	bibi2d 342	. . . . . 6 ⊢ (𝑦 = 𝑢 → ((𝜑 ↔ 𝜓) ↔ (𝜑 ↔ [𝑢 / 𝑦]𝜓)))
20	17, 19	imbi12d 344	. . . . 5 ⊢ (𝑦 = 𝑢 → ((𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓)) ↔ (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))))
21		dfoprab4f.1	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
22	15, 20, 21	chvarfv 2238	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
23	5, 10, 22	chvarfv 2238	. . 3 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
24	23	dfoprab4 8079	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
25		nfv 1912	. . 3 ⊢ Ⅎ𝑡((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)
26		nfv 1912	. . 3 ⊢ Ⅎ𝑢((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)
27		nfv 1912	. . . 4 ⊢ Ⅎ𝑥(𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵)
28	27, 3	nfan 1897	. . 3 ⊢ Ⅎ𝑥((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
29		nfv 1912	. . . 4 ⊢ Ⅎ𝑦(𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵)
30	13	nfsbv 2329	. . . 4 ⊢ Ⅎ𝑦[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
31	29, 30	nfan 1897	. . 3 ⊢ Ⅎ𝑦((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
32		eleq1w 2822	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ 𝐴 ↔ 𝑡 ∈ 𝐴))
33		eleq1w 2822	. . . . 5 ⊢ (𝑦 = 𝑢 → (𝑦 ∈ 𝐵 ↔ 𝑢 ∈ 𝐵))
34	32, 33	bi2anan9 638	. . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵)))
35	18, 8	sylan9bbr 510	. . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
36	34, 35	anbi12d 632	. . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) ↔ ((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
37	25, 26, 28, 31, 36	cbvoprab12 7522	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
38	24, 37	eqtr4i 2766	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  Ⅎwnf 1780  [wsb 2062   ∈ wcel 2106  ⟨cop 4637  {copab 5210   × cxp 5687  {coprab 7432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-oprab 7435  df-1st 8013  df-2nd 8014

This theorem is referenced by: (None)