Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cossid Structured version   Visualization version   GIF version

Theorem cossid 34724
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 2129 . . . 4 (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑧))
2 ideqg 5477 . . . . . . 7 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3389 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
4 ideqg 5477 . . . . . . 7 (𝑧 ∈ V → (𝑥 I 𝑧𝑥 = 𝑧))
54elv 3389 . . . . . 6 (𝑥 I 𝑧𝑥 = 𝑧)
63, 5anbi12i 621 . . . . 5 ((𝑥 I 𝑦𝑥 I 𝑧) ↔ (𝑥 = 𝑦𝑥 = 𝑧))
76exbii 1944 . . . 4 (∃𝑥(𝑥 I 𝑦𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑧))
81, 7bitr4i 270 . . 3 (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧))
98opabbii 4910 . 2 {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧)}
10 df-id 5220 . 2 I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
11 df-coss 34663 . 2 ≀ I = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧)}
129, 10, 113eqtr4ri 2832 1 ≀ I = I
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wex 1875  Vcvv 3385   class class class wbr 4843  {copab 4905   I cid 5219  ccoss 34469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-coss 34663
This theorem is referenced by:  cosscnvid  34725
  Copyright terms: Public domain W3C validator