Theorem bj-dfid2ALT 37729

Description: Alternate version of dfid2 5557. (Contributed by BJ, 9-Nov-2024.) (Proof modification is discouraged.) Use df-id 5555 instead to make the semantics of the construction df-opab 5173 clearer. (New usage is discouraged.)

Assertion

Ref	Expression
bj-dfid2ALT	⊢ I = {⟨𝑥, 𝑥⟩ ∣ ⊤}

Proof of Theorem bj-dfid2ALT

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

Step	Hyp	Ref	Expression
1		df-id 5555	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2		equcomi 2046	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
3	2	opeq2d 4844	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
4	3	eqeq2d 2773	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑥⟩))
5	4	pm5.32ri 585	. . . . . . . . 9 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦))
6	5	exbii 1877	. . . . . . . 8 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦))
7		ax6evr 2044	. . . . . . . . 9 ⊢ ∃𝑦 𝑥 = 𝑦
8		19.42v 1982	. . . . . . . . 9 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 𝑥 = 𝑦))
9	7, 8	mpbiran2 722	. . . . . . . 8 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ 𝑧 = ⟨𝑥, 𝑥⟩)
10	6, 9	bitri 278	. . . . . . 7 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ 𝑧 = ⟨𝑥, 𝑥⟩)
11	10	exbii 1877	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩)
12		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
13	12, 12	opeq12d 4845	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑥⟩ = ⟨𝑢, 𝑢⟩)
14	13	eqeq2d 2773	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑥⟩ ↔ 𝑧 = ⟨𝑢, 𝑢⟩))
15	14	exexw 2082	. . . . . 6 ⊢ (∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩)
16	11, 15	bitri 278	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑥∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩)
17		tru 1573	. . . . . . . 8 ⊢ ⊤
18	17	biantru 538	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑥⟩ ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤))
19	18	exbii 1877	. . . . . 6 ⊢ (∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤))
20	19	exbii 1877	. . . . 5 ⊢ (∃𝑥∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤))
21	16, 20	bitri 278	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤))
22	21	abbii 2829	. . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)} = {𝑧 ∣ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤)}
23		df-opab 5173	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
24		df-opab 5173	. . 3 ⊢ {⟨𝑥, 𝑥⟩ ∣ ⊤} = {𝑧 ∣ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤)}
25	22, 23, 24	3eqtr4i 2795	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑥⟩ ∣ ⊤}
26	1, 25	eqtri 2785	1 ⊢ I = {⟨𝑥, 𝑥⟩ ∣ ⊤}

This proof depends on syntax axioms:   ∧ wa 400   = wceq 1569  ⊤wtru 1570  ∃wex 1808  {cab 2740  ⟨cop 4594  {copab 5172   I cid 5554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173  df-id 5555

This theorem is used by: (None)