Theorem dfoprab4f 7117

Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (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 1830	. . . . 5 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑡, 𝑢⟩
2		dfoprab4f.x	. . . . . 6 ⊢ Ⅎ𝑥𝜑
3		nfs1v 2425	. . . . . 6 ⊢ Ⅎ𝑥[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
4	2, 3	nfbi 1821	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
5	1, 4	nfim 1813	. . . 4 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
6		opeq1 4340	. . . . . 6 ⊢ (𝑥 = 𝑡 → ⟨𝑥, 𝑢⟩ = ⟨𝑡, 𝑢⟩)
7	6	eqeq2d 2620	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑤 = ⟨𝑥, 𝑢⟩ ↔ 𝑤 = ⟨𝑡, 𝑢⟩))
8		sbequ12 2097	. . . . . 6 ⊢ (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
9	8	bibi2d 331	. . . . 5 ⊢ (𝑥 = 𝑡 → ((𝜑 ↔ [𝑢 / 𝑦]𝜓) ↔ (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
10	7, 9	imbi12d 333	. . . 4 ⊢ (𝑥 = 𝑡 → ((𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓)) ↔ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))))
11		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑢⟩
12		dfoprab4f.y	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
13		nfs1v 2425	. . . . . . 7 ⊢ Ⅎ𝑦[𝑢 / 𝑦]𝜓
14	12, 13	nfbi 1821	. . . . . 6 ⊢ Ⅎ𝑦(𝜑 ↔ [𝑢 / 𝑦]𝜓)
15	11, 14	nfim 1813	. . . . 5 ⊢ Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
16		opeq2 4341	. . . . . . 7 ⊢ (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
17	16	eqeq2d 2620	. . . . . 6 ⊢ (𝑦 = 𝑢 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑢⟩))
18		sbequ12 2097	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
19	18	bibi2d 331	. . . . . 6 ⊢ (𝑦 = 𝑢 → ((𝜑 ↔ 𝜓) ↔ (𝜑 ↔ [𝑢 / 𝑦]𝜓)))
20	17, 19	imbi12d 333	. . . . 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	chvar 2250	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
23	5, 10, 22	chvar 2250	. . 3 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
24	23	dfoprab4 7116	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
25		nfv 1830	. . 3 ⊢ Ⅎ𝑡((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)
26		nfv 1830	. . 3 ⊢ Ⅎ𝑢((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)
27		nfv 1830	. . . 4 ⊢ Ⅎ𝑥(𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵)
28	27, 3	nfan 1816	. . 3 ⊢ Ⅎ𝑥((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
29		nfv 1830	. . . 4 ⊢ Ⅎ𝑦(𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵)
30	13	nfsb 2428	. . . 4 ⊢ Ⅎ𝑦[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
31	29, 30	nfan 1816	. . 3 ⊢ Ⅎ𝑦((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
32		eleq1 2676	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ 𝐴 ↔ 𝑡 ∈ 𝐴))
33		eleq1 2676	. . . . 5 ⊢ (𝑦 = 𝑢 → (𝑦 ∈ 𝐵 ↔ 𝑢 ∈ 𝐵))
34	32, 33	bi2anan9 913	. . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵)))
35	18, 8	sylan9bbr 733	. . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
36	34, 35	anbi12d 743	. . 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 6627	. 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 2635	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699  [wsb 1867   ∈ wcel 1977  ⟨cop 4131  {copab 4642   × cxp 5036  {coprab 6550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-oprab 6553  df-1st 7059  df-2nd 7060

This theorem is referenced by: (None)