Theorem cossid 38471

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 2029	. . . 4 ⊢ (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑧))
2		ideqg 5815	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
3	2	elv 3452	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4		ideqg 5815	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑥 I 𝑧 ↔ 𝑥 = 𝑧))
5	4	elv 3452	. . . . . 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 1848	. . . 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 5174	. 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)}
10		df-id 5533	. 2 ⊢ I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
11		df-coss 38402	. 2 ⊢ ≀ I = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦 ∧ 𝑥 I 𝑧)}
12	9, 10, 11	3eqtr4ri 2763	1 ⊢ ≀ I = I

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779  Vcvv 3447   class class class wbr 5107  {copab 5169   I cid 5532   ≀ ccoss 38169

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-coss 38402

This theorem is referenced by:  cosscnvid  38472  eqvrelid  38781