Theorem isexid2 37817

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 37815	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
3		elin 3992	. . . . . . 7 ⊢ (𝐺 ∈ (Magma ∩ ExId ) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ ExId ))
4		eqid 2740	. . . . . . . . . . 11 ⊢ dom dom 𝐺 = dom dom 𝐺
5	4	isexid 37809	. . . . . . . . . 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 2752	. . . . . . 7 ⊢ (ran 𝐺 = dom dom 𝐺 → (𝑋 = ran 𝐺 ↔ 𝑋 = dom dom 𝐺))
11		raleq 3331	. . . . . . . 8 ⊢ (ran 𝐺 = dom dom 𝐺 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
12	11	rexeqbi1dv 3347	. . . . . . 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 3331	. . . 4 ⊢ (𝑋 = ran 𝐺 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
18	17	rexeqbi1dv 3347	. . 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 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∩ cin 3975  dom cdm 5700  ran crn 5701  (class class class)co 7450   ExId cexid 37806  Magmacmagm 37810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fo 6581  df-fv 6583  df-ov 7453  df-exid 37807  df-mgmOLD 37811

This theorem is referenced by:  exidu1  37818