Theorem dfid2 5491

Description: Alternate definition of the identity relation. Instance of dfid3 5492 not requiring auxiliary axioms. (Contributed by NM, 15-Mar-2007.) Reduce axiom usage. (Revised by Gino Giotto, 4-Nov-2024.) (Proof shortened by BJ, 5-Nov-2024.)

Use df-id 5489 instead to make the semantics of the constructor df-opab 5137 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 5489	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2		equcomi 2020	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
3	2	opeq2d 4811	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
4	3	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑥⟩))
5	4	pm5.32ri 576	. . . . . . . . 9 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦))
6	5	exbii 1850	. . . . . . . 8 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦))
7		ax6evr 2018	. . . . . . . . 9 ⊢ ∃𝑦 𝑥 = 𝑦
8		19.42v 1957	. . . . . . . . 9 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 𝑥 = 𝑦))
9	7, 8	mpbiran2 707	. . . . . . . 8 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ 𝑧 = ⟨𝑥, 𝑥⟩)
10	6, 9	bitri 274	. . . . . . 7 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ 𝑧 = ⟨𝑥, 𝑥⟩)
11		eqidd 2739	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑥⟩ → 𝑥 = 𝑥)
12	11	pm4.71i 560	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑥⟩ ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
13	10, 12	bitri 274	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
14	13	exbii 1850	. . . . 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 4812	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑥⟩ = ⟨𝑢, 𝑢⟩)
17	16	eqeq2d 2749	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑥⟩ ↔ 𝑧 = ⟨𝑢, 𝑢⟩))
18	15, 15	eqeq12d 2754	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑥 ↔ 𝑢 = 𝑢))
19	17, 18	anbi12d 631	. . . . . 6 ⊢ (𝑥 = 𝑢 → ((𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑧 = ⟨𝑢, 𝑢⟩ ∧ 𝑢 = 𝑢)))
20	19	exexw 2054	. . . . 5 ⊢ (∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
21	14, 20	bitri 274	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
22	21	abbii 2808	. . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)} = {𝑧 ∣ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥)}
23		df-opab 5137	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
24		df-opab 5137	. . 3 ⊢ {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥} = {𝑧 ∣ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥)}
25	22, 23, 24	3eqtr4i 2776	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}
26	1, 25	eqtri 2766	1 ⊢ I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}

Syntax hints:   ∧ wa 396   = wceq 1539  ∃wex 1782  {cab 2715  ⟨cop 4567  {copab 5136   I cid 5488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-id 5489

This theorem is referenced by:  fsplitOLD  7958