Theorem iorlid 37865

Description: A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
iorlid.1	⊢ 𝑋 = ran 𝐺
iorlid.2	⊢ 𝑈 = (GId‘𝐺)

Assertion

Ref	Expression
iorlid	⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋)

Proof of Theorem iorlid

Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iorlid.1	. . 3 ⊢ 𝑋 = ran 𝐺
2		iorlid.2	. . 3 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	idrval 37864	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
4	1	exidu1 37863	. . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
5		riotacl 7405	. . 3 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ 𝑋)
6	4, 5	syl 17	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ 𝑋)
7	3, 6	eqeltrd 2841	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃!wreu 3378   ∩ cin 3950  ran crn 5686  ‘cfv 6561  ℩crio 7387  (class class class)co 7431  GIdcgi 30509   ExId cexid 37851  Magmacmagm 37855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-riota 7388  df-ov 7434  df-gid 30513  df-exid 37852  df-mgmOLD 37856

This theorem is referenced by:  cmpidelt  37866  rngo1cl  37946