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 38474
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 2028 . . . 4 (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑧))
2 ideqg 5860 . . . . . . 7 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3484 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
4 ideqg 5860 . . . . . . 7 (𝑧 ∈ V → (𝑥 I 𝑧𝑥 = 𝑧))
54elv 3484 . . . . . 6 (𝑥 I 𝑧𝑥 = 𝑧)
63, 5anbi12i 628 . . . . 5 ((𝑥 I 𝑦𝑥 I 𝑧) ↔ (𝑥 = 𝑦𝑥 = 𝑧))
76exbii 1848 . . . 4 (∃𝑥(𝑥 I 𝑦𝑥 I 𝑧) ↔ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑧))
81, 7bitr4i 278 . . 3 (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧))
98opabbii 5208 . 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 38405 . 2 ≀ I = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧)}
129, 10, 113eqtr4ri 2775 1 ≀ I = I
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  Vcvv 3479   class class class wbr 5141  {copab 5203   I cid 5575  ccoss 38175
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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-id 5576  df-xp 5689  df-rel 5690  df-coss 38405
This theorem is referenced by:  cosscnvid  38475  eqvrelid  38783
  Copyright terms: Public domain W3C validator