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Theorem gidval 30335
Description: The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
gidval.1 𝑋 = ran 𝐺
Assertion
Ref Expression
gidval (𝐺𝑉 → (GId‘𝐺) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
Distinct variable groups:   𝑥,𝑢,𝐺   𝑢,𝑋,𝑥
Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem gidval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 elex 3490 . 2 (𝐺𝑉𝐺 ∈ V)
2 rneq 5938 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
3 gidval.1 . . . . 5 𝑋 = ran 𝐺
42, 3eqtr4di 2786 . . . 4 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
5 oveq 7426 . . . . . . 7 (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
65eqeq1d 2730 . . . . . 6 (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
7 oveq 7426 . . . . . . 7 (𝑔 = 𝐺 → (𝑥𝑔𝑢) = (𝑥𝐺𝑢))
87eqeq1d 2730 . . . . . 6 (𝑔 = 𝐺 → ((𝑥𝑔𝑢) = 𝑥 ↔ (𝑥𝐺𝑢) = 𝑥))
96, 8anbi12d 631 . . . . 5 (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
104, 9raleqbidv 3339 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
114, 10riotaeqbidv 7379 . . 3 (𝑔 = 𝐺 → (𝑢 ∈ ran 𝑔𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
12 df-gid 30317 . . 3 GId = (𝑔 ∈ V ↦ (𝑢 ∈ ran 𝑔𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
13 riotaex 7380 . . 3 (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ V
1411, 12, 13fvmpt 7005 . 2 (𝐺 ∈ V → (GId‘𝐺) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
151, 14syl 17 1 (𝐺𝑉 → (GId‘𝐺) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3058  Vcvv 3471  ran crn 5679  cfv 6548  crio 7375  (class class class)co 7420  GIdcgi 30313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-riota 7376  df-ov 7423  df-gid 30317
This theorem is referenced by:  grpoidval  30336  idrval  37330  exidresid  37352
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