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

Theorem cosseq 34542
Description: Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
Assertion
Ref Expression
cosseq (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)

Proof of Theorem cosseq
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4811 . . . . 5 (𝐴 = 𝐵 → (𝑢𝐴𝑥𝑢𝐵𝑥))
2 breq 4811 . . . . 5 (𝐴 = 𝐵 → (𝑢𝐴𝑦𝑢𝐵𝑦))
31, 2anbi12d 624 . . . 4 (𝐴 = 𝐵 → ((𝑢𝐴𝑥𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥𝑢𝐵𝑦)))
43exbidv 2016 . . 3 (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)))
54opabbidv 4875 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)})
6 df-coss 34530 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦)}
7 df-coss 34530 . 2 𝐵 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)}
85, 6, 73eqtr4g 2824 1 (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wex 1874   class class class wbr 4809  {copab 4871  ccoss 34336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-br 4810  df-opab 4872  df-coss 34530
This theorem is referenced by:  cosseqi  34543  cosseqd  34544
  Copyright terms: Public domain W3C validator