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Theorem isexid 33776
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 5356 . . . . 5 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
21dmeqd 5358 . . . 4 (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
3 isexid.1 . . . 4 𝑋 = dom dom 𝐺
42, 3syl6eqr 2703 . . 3 (𝑔 = 𝐺 → dom dom 𝑔 = 𝑋)
5 oveq 6696 . . . . . 6 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
65eqeq1d 2653 . . . . 5 (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = 𝑦 ↔ (𝑥𝐺𝑦) = 𝑦))
7 oveq 6696 . . . . . 6 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
87eqeq1d 2653 . . . . 5 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑦 ↔ (𝑦𝐺𝑥) = 𝑦))
96, 8anbi12d 747 . . . 4 (𝑔 = 𝐺 → (((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
104, 9raleqbidv 3182 . . 3 (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∀𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
114, 10rexeqbidv 3183 . 2 (𝑔 = 𝐺 → (∃𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
12 df-exid 33774 . 2 ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
1311, 12elab2g 3385 1 (𝐺𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  dom cdm 5143  (class class class)co 6690   ExId cexid 33773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-dm 5153  df-iota 5889  df-fv 5934  df-ov 6693  df-exid 33774
This theorem is referenced by:  opidonOLD  33781  isexid2  33784  ismndo  33801  exidres  33807
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