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Theorem isexid 37226
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 5896 . . . . 5 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
21dmeqd 5898 . . . 4 (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
3 isexid.1 . . . 4 𝑋 = dom dom 𝐺
42, 3eqtr4di 2784 . . 3 (𝑔 = 𝐺 → dom dom 𝑔 = 𝑋)
5 oveq 7410 . . . . . 6 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
65eqeq1d 2728 . . . . 5 (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = 𝑦 ↔ (𝑥𝐺𝑦) = 𝑦))
7 oveq 7410 . . . . . 6 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
87eqeq1d 2728 . . . . 5 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑦 ↔ (𝑦𝐺𝑥) = 𝑦))
96, 8anbi12d 630 . . . 4 (𝑔 = 𝐺 → (((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
104, 9raleqbidv 3336 . . 3 (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∀𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
114, 10rexeqbidv 3337 . 2 (𝑔 = 𝐺 → (∃𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
12 df-exid 37224 . 2 ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
1311, 12elab2g 3665 1 (𝐺𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  dom cdm 5669  (class class class)co 7404   ExId cexid 37223
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
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-uni 4903  df-br 5142  df-dm 5679  df-iota 6488  df-fv 6544  df-ov 7407  df-exid 37224
This theorem is referenced by:  opidonOLD  37231  isexid2  37234  ismndo  37251  exidres  37257
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