Theorem cossid 38191

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 2025	. . . 4 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧))
2		ideqg 5850	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
3	2	elv 3468	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4		ideqg 5850	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑥 I 𝑧 ↔ 𝑥 = 𝑧))
5	4	elv 3468	. . . . . 6 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
6	3, 5	anbi12i 626	. . . . 5 ⊢ ((𝑥 I 𝑦 ∧ 𝑥 I 𝑧) ↔ (𝑥 = 𝑦 ∧ 𝑥 = 𝑧))
7	6	exbii 1843	. . . 4 ⊢ (∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧))
8	1, 7	bitr4i 277	. . 3 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧))
9	8	opabbii 5212	. 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)}
10		df-id 5572	. 2 ⊢ I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
11		df-coss 38122	. 2 ⊢ ≀ I = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)}
12	9, 10, 11	3eqtr4ri 2765	1 ⊢ ≀ I = I

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1534  ∃wex 1774  Vcvv 3462   class class class wbr 5145  {copab 5207   I cid 5571   ≀ ccoss 37889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-coss 38122

This theorem is referenced by:  cosscnvid  38192  eqvrelid  38500