Theorem cossid 38523

Description: Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.)

Assertion

Ref	Expression
cossid	⊢ ≀ I = I

Proof of Theorem cossid

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

Step	Hyp	Ref	Expression
1		equvinv 2030	. . . 4 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧))
2		ideqg 5791	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
3	2	elv 3441	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4		ideqg 5791	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑥 I 𝑧 ↔ 𝑥 = 𝑧))
5	4	elv 3441	. . . . . 6 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
6	3, 5	anbi12i 628	. . . . 5 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 I 𝑧) ↔ (𝑥 = 𝑦 ∧ 𝑥 = 𝑧))
7	6	exbii 1849	. . . 4 ⊢ (∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧))
8	1, 7	bitr4i 278	. . 3 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧))
9	8	opabbii 5158	. 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)}
10		df-id 5511	. 2 ⊢ I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
11		df-coss 38454	. 2 ⊢ ≀ I = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)}
12	9, 10, 11	3eqtr4ri 2765	1 ⊢ ≀ I = I

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780  Vcvv 3436   class class class wbr 5091  {copab 5153   I cid 5510   ≀ ccoss 38221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-coss 38454

This theorem is referenced by:  cosscnvid  38524  eqvrelid  38833