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 37861
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 2024 . . . 4 (𝑦 = 𝑧 ↔ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑧))
2 ideqg 5844 . . . . . . 7 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3474 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
4 ideqg 5844 . . . . . . 7 (𝑧 ∈ V → (𝑥 I 𝑧𝑥 = 𝑧))
54elv 3474 . . . . . 6 (𝑥 I 𝑧𝑥 = 𝑧)
63, 5anbi12i 626 . . . . 5 ((𝑥 I 𝑦𝑥 I 𝑧) ↔ (𝑥 = 𝑦𝑥 = 𝑧))
76exbii 1842 . . . 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 5567 . 2 I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
11 df-coss 37792 . 2 ≀ I = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥 I 𝑦𝑥 I 𝑧)}
129, 10, 113eqtr4ri 2765 1 ≀ I = I
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wex 1773  Vcvv 3468   class class class wbr 5141  {copab 5203   I cid 5566  ccoss 37554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-coss 37792
This theorem is referenced by:  cosscnvid  37862  eqvrelid  38170
  Copyright terms: Public domain W3C validator