Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isexid Structured version   Visualization version   GIF version

Theorem isexid 34604
Description: The predicate 𝐺 has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
isexid.1 𝑋 = dom dom 𝐺
Assertion
Ref Expression
isexid (𝐺𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
Distinct variable groups:   𝑥,𝐺,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem isexid
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 dmeq 5618 . . . . 5 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
21dmeqd 5620 . . . 4 (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
3 isexid.1 . . . 4 𝑋 = dom dom 𝐺
42, 3syl6eqr 2825 . . 3 (𝑔 = 𝐺 → dom dom 𝑔 = 𝑋)
5 oveq 6980 . . . . . 6 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
65eqeq1d 2773 . . . . 5 (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = 𝑦 ↔ (𝑥𝐺𝑦) = 𝑦))
7 oveq 6980 . . . . . 6 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
87eqeq1d 2773 . . . . 5 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑦 ↔ (𝑦𝐺𝑥) = 𝑦))
96, 8anbi12d 622 . . . 4 (𝑔 = 𝐺 → (((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
104, 9raleqbidv 3334 . . 3 (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∀𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
114, 10rexeqbidv 3335 . 2 (𝑔 = 𝐺 → (∃𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
12 df-exid 34602 . 2 ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
1311, 12elab2g 3577 1 (𝐺𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  wral 3081  wrex 3082  dom cdm 5403  (class class class)co 6974   ExId cexid 34601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-dm 5413  df-iota 6149  df-fv 6193  df-ov 6977  df-exid 34602
This theorem is referenced by:  opidonOLD  34609  isexid2  34612  ismndo  34629  exidres  34635
  Copyright terms: Public domain W3C validator