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Theorem cossid 37952
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 5854 . . . . . . 7 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3477 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
4 ideqg 5854 . . . . . . 7 (𝑧 ∈ V → (𝑥 I 𝑧𝑥 = 𝑧))
54elv 3477 . . . . . 6 (𝑥 I 𝑧𝑥 = 𝑧)
63, 5anbi12i 627 . . . . 5 ((𝑥 I 𝑦𝑥 I 𝑧) ↔ (𝑥 = 𝑦𝑥 = 𝑧))
76exbii 1843 . . . 4 (∃𝑥(𝑥 I 𝑦𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑧))
81, 7bitr4i 278 . . 3 (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧))
98opabbii 5215 . 2 {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧)}
10 df-id 5576 . 2 I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
11 df-coss 37883 . 2 ≀ I = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧)}
129, 10, 113eqtr4ri 2767 1 ≀ I = I
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wex 1774  Vcvv 3471   class class class wbr 5148  {copab 5210   I cid 5575  ccoss 37648
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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-coss 37883
This theorem is referenced by:  cosscnvid  37953  eqvrelid  38261
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