Theorem gidval 30598

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	⊢ (𝐺 ∈ 𝑉 → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((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.

Step	Hyp	Ref	Expression
1		elex 3451	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		rneq 5885	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
3		gidval.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
4	2, 3	eqtr4di 2790	. . . 4 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
5		oveq 7366	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
6	5	eqeq1d 2739	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
7		oveq 7366	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑢) = (𝑥𝐺𝑢))
8	7	eqeq1d 2739	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑢) = 𝑥 ↔ (𝑥𝐺𝑢) = 𝑥))
9	6, 8	anbi12d 633	. . . . 5 ⊢ (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
10	4, 9	raleqbidv 3312	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
11	4, 10	riotaeqbidv 7320	. . 3 ⊢ (𝑔 = 𝐺 → (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
12		df-gid 30580	. . 3 ⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
13		riotaex 7321	. . 3 ⊢ (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ V
14	11, 12, 13	fvmpt 6941	. 2 ⊢ (𝐺 ∈ V → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
15	1, 14	syl 17	1 ⊢ (𝐺 ∈ 𝑉 → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ran crn 5625  ‘cfv 6492  ℩crio 7316  (class class class)co 7360  GIdcgi 30576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-riota 7317  df-ov 7363  df-gid 30580

This theorem is referenced by:  grpoidval  30599  idrval  38192  exidresid  38214