Theorem eloprab1st2nd 49796

Description: Reconstruction of a nested ordered pair in terms of its ordered pair components. (Contributed by Zhi Wang, 27-Oct-2025.)

Assertion

Ref	Expression
eloprab1st2nd	⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝐴 = ⟨⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩, (2nd ‘𝐴)⟩)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑧,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem eloprab1st2nd

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2764	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
2	1	anbi1d 643	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3	2	3exbidv 1958	. . . 4 ⊢ (𝑤 = 𝐴 → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
4		df-oprab 7417	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
5	3, 4	elab2g 3634	. . 3 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
6	5	ibi 270	. 2 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7		id 23	. . . . . 6 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
8		opex 5439	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
9		vex 3454	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
10	8, 9	op1std 7996	. . . . . . . . . 10 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘𝐴) = ⟨𝑥, 𝑦⟩)
11	10	fveq2d 6882	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘(1st ‘𝐴)) = (1st ‘⟨𝑥, 𝑦⟩))
12		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
13		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
14	12, 13	op1st 7994	. . . . . . . . 9 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
15	11, 14	eqtr2di 2812	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑥 = (1st ‘(1st ‘𝐴)))
16	10	fveq2d 6882	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘⟨𝑥, 𝑦⟩))
17	12, 13	op2nd 7995	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
18	16, 17	eqtr2di 2812	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑦 = (2nd ‘(1st ‘𝐴)))
19	15, 18	opeq12d 4841	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ⟨𝑥, 𝑦⟩ = ⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩)
20	8, 9	op2ndd 7997	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘𝐴) = 𝑧)
21	20	eqcomd 2766	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑧 = (2nd ‘𝐴))
22	19, 21	opeq12d 4841	. . . . . 6 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩, (2nd ‘𝐴)⟩)
23	7, 22	eqtrd 2795	. . . . 5 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝐴 = ⟨⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩, (2nd ‘𝐴)⟩)
24	23	adantr 486	. . . 4 ⊢ ((𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩, (2nd ‘𝐴)⟩)
25	24	exlimiv 1963	. . 3 ⊢ (∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩, (2nd ‘𝐴)⟩)
26	25	exlimivv 1965	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩, (2nd ‘𝐴)⟩)
27	6, 26	syl 18	1 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝐴 = ⟨⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩, (2nd ‘𝐴)⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ⟨cop 4590  ‘cfv 6533  {coprab 7414  1^st c1st 7984  2^nd c2nd 7985

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fv 6541  df-oprab 7417  df-1st 7986  df-2nd 7987

This theorem is used by:  sectpropdlem  49962  invpropdlem  49964  isopropdlem  49966