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Theorem gidval 30270
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 3487 . 2 (𝐺𝑉𝐺 ∈ V)
2 rneq 5928 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
3 gidval.1 . . . . 5 𝑋 = ran 𝐺
42, 3eqtr4di 2784 . . . 4 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
5 oveq 7410 . . . . . . 7 (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
65eqeq1d 2728 . . . . . 6 (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
7 oveq 7410 . . . . . . 7 (𝑔 = 𝐺 → (𝑥𝑔𝑢) = (𝑥𝐺𝑢))
87eqeq1d 2728 . . . . . 6 (𝑔 = 𝐺 → ((𝑥𝑔𝑢) = 𝑥 ↔ (𝑥𝐺𝑢) = 𝑥))
96, 8anbi12d 630 . . . . 5 (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
104, 9raleqbidv 3336 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
114, 10riotaeqbidv 7363 . . 3 (𝑔 = 𝐺 → (𝑢 ∈ ran 𝑔𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
12 df-gid 30252 . . 3 GId = (𝑔 ∈ V ↦ (𝑢 ∈ ran 𝑔𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
13 riotaex 7364 . . 3 (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ V
1411, 12, 13fvmpt 6991 . 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 1533  wcel 2098  wral 3055  Vcvv 3468  ran crn 5670  cfv 6536  crio 7359  (class class class)co 7404  GIdcgi 30248
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  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-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fv 6544  df-riota 7360  df-ov 7407  df-gid 30252
This theorem is referenced by:  grpoidval  30271  idrval  37236  exidresid  37258
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