Theorem isexid2 37995

Description: If 𝐺 ∈ (Magma ∩ ExId ), then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
isexid2.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
isexid2	⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))

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

Proof of Theorem isexid2

Step	Hyp	Ref	Expression
1		isexid2.1	. 2 ⊢ 𝑋 = ran 𝐺
2		rngopidOLD 37993	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
3		elin 3915	. . . . . . 7 ⊢ (𝐺 ∈ (Magma ∩ ExId ) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ ExId ))
4		eqid 2734	. . . . . . . . . . 11 ⊢ dom dom 𝐺 = dom dom 𝐺
5	4	isexid 37987	. . . . . . . . . 10 ⊢ (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
6	5	ibi 267	. . . . . . . . 9 ⊢ (𝐺 ∈ ExId → ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
7	6	a1d 25	. . . . . . . 8 ⊢ (𝐺 ∈ ExId → (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
8	7	adantl 481	. . . . . . 7 ⊢ ((𝐺 ∈ Magma ∧ 𝐺 ∈ ExId ) → (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
9	3, 8	sylbi 217	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
10		eqeq2 2746	. . . . . . 7 ⊢ (ran 𝐺 = dom dom 𝐺 → (𝑋 = ran 𝐺 ↔ 𝑋 = dom dom 𝐺))
11		raleq 3291	. . . . . . . 8 ⊢ (ran 𝐺 = dom dom 𝐺 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
12	11	rexeqbi1dv 3307	. . . . . . 7 ⊢ (ran 𝐺 = dom dom 𝐺 → (∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
13	10, 12	imbi12d 344	. . . . . 6 ⊢ (ran 𝐺 = dom dom 𝐺 → ((𝑋 = ran 𝐺 → ∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ↔ (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))))
14	9, 13	imbitrrid 246	. . . . 5 ⊢ (ran 𝐺 = dom dom 𝐺 → (𝐺 ∈ (Magma ∩ ExId ) → (𝑋 = ran 𝐺 → ∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))))
15	2, 14	mpcom 38	. . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝑋 = ran 𝐺 → ∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
16	15	com12 32	. . 3 ⊢ (𝑋 = ran 𝐺 → (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
17		raleq 3291	. . . 4 ⊢ (𝑋 = ran 𝐺 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
18	17	rexeqbi1dv 3307	. . 3 ⊢ (𝑋 = ran 𝐺 → (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
19	16, 18	sylibrd 259	. 2 ⊢ (𝑋 = ran 𝐺 → (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
20	1, 19	ax-mp 5	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058   ∩ cin 3898  dom cdm 5622  ran crn 5623  (class class class)co 7356   ExId cexid 37984  Magmacmagm 37988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-ov 7359  df-exid 37985  df-mgmOLD 37989

This theorem is referenced by:  exidu1  37996