Theorem dfid2 5546

Description: Alternate definition of the identity relation. Instance of dfid3 5547 not requiring auxiliary axioms. (Contributed by NM, 15-Mar-2007.) Reduce axiom usage. (Revised by GG, 4-Nov-2024.) (Proof shortened by BJ, 5-Nov-2024.)

Use df-id 5544 instead to make the semantics of the constructor df-opab 5165 clearer. (New usage is discouraged.)

Assertion

Ref	Expression
dfid2	⊢ I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}

Proof of Theorem dfid2

Dummy variables 𝑦 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-id 5544	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2		equcomi 2039	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
3	2	opeq2d 4840	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
4	3	eqeq2d 2775	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑥⟩))
5	4	pm5.32ri 583	. . . . . . . . 9 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦))
6	5	exbii 1870	. . . . . . . 8 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦))
7		ax6evr 2037	. . . . . . . . 9 ⊢ ∃𝑦 𝑥 = 𝑦
8		19.42v 1975	. . . . . . . . 9 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 𝑥 = 𝑦))
9	7, 8	mpbiran2 720	. . . . . . . 8 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ 𝑧 = ⟨𝑥, 𝑥⟩)
10	6, 9	bitri 277	. . . . . . 7 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ 𝑧 = ⟨𝑥, 𝑥⟩)
11		eqidd 2765	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑥⟩ → 𝑥 = 𝑥)
12	11	pm4.71i 567	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑥⟩ ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
13	10, 12	bitri 277	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
14	13	exbii 1870	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
15		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
16	15, 15	opeq12d 4841	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑥⟩ = ⟨𝑢, 𝑢⟩)
17	16	eqeq2d 2775	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑥⟩ ↔ 𝑧 = ⟨𝑢, 𝑢⟩))
18	15, 15	eqeq12d 2780	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑥 ↔ 𝑢 = 𝑢))
19	17, 18	anbi12d 641	. . . . . 6 ⊢ (𝑥 = 𝑢 → ((𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑧 = ⟨𝑢, 𝑢⟩ ∧ 𝑢 = 𝑢)))
20	19	exexw 2075	. . . . 5 ⊢ (∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
21	14, 20	bitri 277	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
22	21	abbii 2831	. . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)} = {𝑧 ∣ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥)}
23		df-opab 5165	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
24		df-opab 5165	. . 3 ⊢ {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥} = {𝑧 ∣ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥)}
25	22, 23, 24	3eqtr4i 2797	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}
26	1, 25	eqtri 2787	1 ⊢ I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}

Syntax hints:   ∧ wa 399   = wceq 1562  ∃wex 1801  {cab 2742  ⟨cop 4590  {copab 5164   I cid 5543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5165  df-id 5544

This theorem is referenced by: (None)