Theorem bj-dfid2ALT 37048

Description: Alternate version of dfid2 5537. (Contributed by BJ, 9-Nov-2024.) (Proof modification is discouraged.) Use df-id 5535 instead to make the semantics of the construction df-opab 5172 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 5535	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2		equcomi 2017	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
3	2	opeq2d 4846	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
4	3	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑥⟩))
5	4	pm5.32ri 575	. . . . . . . . 9 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦))
6	5	exbii 1848	. . . . . . . 8 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦))
7		ax6evr 2015	. . . . . . . . 9 ⊢ ∃𝑦 𝑥 = 𝑦
8		19.42v 1953	. . . . . . . . 9 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 𝑥 = 𝑦))
9	7, 8	mpbiran2 710	. . . . . . . 8 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ 𝑧 = ⟨𝑥, 𝑥⟩)
10	6, 9	bitri 275	. . . . . . 7 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ 𝑧 = ⟨𝑥, 𝑥⟩)
11	10	exbii 1848	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩)
12		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
13	12, 12	opeq12d 4847	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑥⟩ = ⟨𝑢, 𝑢⟩)
14	13	eqeq2d 2741	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑥⟩ ↔ 𝑧 = ⟨𝑢, 𝑢⟩))
15	14	exexw 2052	. . . . . 6 ⊢ (∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩)
16	11, 15	bitri 275	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑥∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩)
17		tru 1544	. . . . . . . 8 ⊢ ⊤
18	17	biantru 529	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑥⟩ ↔ (𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤))
19	18	exbii 1848	. . . . . 6 ⊢ (∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤))
20	19	exbii 1848	. . . . 5 ⊢ (∃𝑥∃𝑥 𝑧 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤))
21	16, 20	bitri 275	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤))
22	21	abbii 2797	. . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)} = {𝑧 ∣ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤)}
23		df-opab 5172	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
24		df-opab 5172	. . 3 ⊢ {⟨𝑥, 𝑥⟩ ∣ ⊤} = {𝑧 ∣ ∃𝑥∃𝑥(𝑧 = ⟨𝑥, 𝑥⟩ ∧ ⊤)}
25	22, 23, 24	3eqtr4i 2763	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑥⟩ ∣ ⊤}
26	1, 25	eqtri 2753	1 ⊢ I = {⟨𝑥, 𝑥⟩ ∣ ⊤}

Syntax hints:   ∧ wa 395   = wceq 1540  ⊤wtru 1541  ∃wex 1779  {cab 2708  ⟨cop 4597  {copab 5171   I cid 5534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5172  df-id 5535

This theorem is referenced by: (None)