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Theorem gidval 29496
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 3462 . 2 (𝐺𝑉𝐺 ∈ V)
2 rneq 5892 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
3 gidval.1 . . . . 5 𝑋 = ran 𝐺
42, 3eqtr4di 2791 . . . 4 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
5 oveq 7364 . . . . . . 7 (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
65eqeq1d 2735 . . . . . 6 (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
7 oveq 7364 . . . . . . 7 (𝑔 = 𝐺 → (𝑥𝑔𝑢) = (𝑥𝐺𝑢))
87eqeq1d 2735 . . . . . 6 (𝑔 = 𝐺 → ((𝑥𝑔𝑢) = 𝑥 ↔ (𝑥𝐺𝑢) = 𝑥))
96, 8anbi12d 632 . . . . 5 (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
104, 9raleqbidv 3318 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
114, 10riotaeqbidv 7317 . . 3 (𝑔 = 𝐺 → (𝑢 ∈ ran 𝑔𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
12 df-gid 29478 . . 3 GId = (𝑔 ∈ V ↦ (𝑢 ∈ ran 𝑔𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
13 riotaex 7318 . . 3 (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ V
1411, 12, 13fvmpt 6949 . 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 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3444  ran crn 5635  cfv 6497  crio 7313  (class class class)co 7358  GIdcgi 29474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-riota 7314  df-ov 7361  df-gid 29478
This theorem is referenced by:  grpoidval  29497  idrval  36362  exidresid  36384
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