Theorem iorlid 37859

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 37858	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
4	1	exidu1 37857	. . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
5		riotacl 7364	. . 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 2829	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃!wreu 3354   ∩ cin 3916  ran crn 5642  ‘cfv 6514  ℩crio 7346  (class class class)co 7390  GIdcgi 30426   ExId cexid 37845  Magmacmagm 37849

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-riota 7347  df-ov 7393  df-gid 30430  df-exid 37846  df-mgmOLD 37850

This theorem is referenced by:  cmpidelt  37860  rngo1cl  37940