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Theorem cossid 38461
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.
StepHypRef Expression
1 equvinv 2025 . . . 4 (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑧))
2 ideqg 5864 . . . . . . 7 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3482 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
4 ideqg 5864 . . . . . . 7 (𝑧 ∈ V → (𝑥 I 𝑧𝑥 = 𝑧))
54elv 3482 . . . . . 6 (𝑥 I 𝑧𝑥 = 𝑧)
63, 5anbi12i 628 . . . . 5 ((𝑥 I 𝑦𝑥 I 𝑧) ↔ (𝑥 = 𝑦𝑥 = 𝑧))
76exbii 1844 . . . 4 (∃𝑥(𝑥 I 𝑦𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑧))
81, 7bitr4i 278 . . 3 (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧))
98opabbii 5214 . 2 {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧)}
10 df-id 5582 . 2 I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
11 df-coss 38392 . 2 ≀ I = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧)}
129, 10, 113eqtr4ri 2773 1 ≀ I = I
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wex 1775  Vcvv 3477   class class class wbr 5147  {copab 5209   I cid 5581  ccoss 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-coss 38392
This theorem is referenced by:  cosscnvid  38462  eqvrelid  38770
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