Theorem exidu1 37857

Description: Uniqueness of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
exidu1.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
exidu1	⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))

Distinct variable groups:   𝑢,𝐺,𝑥   𝑢,𝑋,𝑥

Proof of Theorem exidu1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exidu1.1	. . 3 ⊢ 𝑋 = ran 𝐺
2	1	isexid2 37856	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
3		simpl 482	. . . . . . 7 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
4	3	ralimi 3067	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
5		oveq2 7398	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
6		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
7	5, 6	eqeq12d 2746	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
8	7	rspcv 3587	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
9	4, 8	syl5 34	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑦) = 𝑦))
10		simpr 484	. . . . . . 7 ⊢ (((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑥𝐺𝑦) = 𝑥)
11	10	ralimi 3067	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑥𝐺𝑦) = 𝑥)
12		oveq1 7397	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥𝐺𝑦) = (𝑢𝐺𝑦))
13		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
14	12, 13	eqeq12d 2746	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ((𝑥𝐺𝑦) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑢))
15	14	rspcv 3587	. . . . . 6 ⊢ (𝑢 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑥𝐺𝑦) = 𝑥 → (𝑢𝐺𝑦) = 𝑢))
16	11, 15	syl5 34	. . . . 5 ⊢ (𝑢 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑢𝐺𝑦) = 𝑢))
17	9, 16	im2anan9r 621	. . . 4 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → ((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢)))
18		eqtr2 2751	. . . . 5 ⊢ (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑦 = 𝑢)
19	18	equcomd 2019	. . . 4 ⊢ (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑢 = 𝑦)
20	17, 19	syl6 35	. . 3 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦))
21	20	rgen2 3178	. 2 ⊢ ∀𝑢 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦)
22		oveq1 7397	. . . . 5 ⊢ (𝑢 = 𝑦 → (𝑢𝐺𝑥) = (𝑦𝐺𝑥))
23	22	eqeq1d 2732	. . . 4 ⊢ (𝑢 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑦𝐺𝑥) = 𝑥))
24	23	ovanraleqv 7414	. . 3 ⊢ (𝑢 = 𝑦 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)))
25	24	reu4 3705	. 2 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑢 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦)))
26	2, 21, 25	sylanblrc 590	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  ∃!wreu 3354   ∩ cin 3916  ran crn 5642  (class class class)co 7390   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-ov 7393  df-exid 37846  df-mgmOLD 37850

This theorem is referenced by:  iorlid  37859  cmpidelt  37860  exidresid  37880