Theorem isexid2 36041

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 36039	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
3		elin 3905	. . . . . . 7 ⊢ (𝐺 ∈ (Magma ∩ ExId ) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ ExId ))
4		eqid 2733	. . . . . . . . . . 11 ⊢ dom dom 𝐺 = dom dom 𝐺
5	4	isexid 36033	. . . . . . . . . 10 ⊢ (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
6	5	ibi 266	. . . . . . . . 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 216	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
10		eqeq2 2745	. . . . . . 7 ⊢ (ran 𝐺 = dom dom 𝐺 → (𝑋 = ran 𝐺 ↔ 𝑋 = dom dom 𝐺))
11		raleq 3344	. . . . . . . 8 ⊢ (ran 𝐺 = dom dom 𝐺 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
12	11	rexeqbi1dv 3343	. . . . . . 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	syl5ibr 245	. . . . 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 3344	. . . 4 ⊢ (𝑋 = ran 𝐺 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
18	17	rexeqbi1dv 3343	. . 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 258	. 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 2101  ∀wral 3059  ∃wrex 3068   ∩ cin 3888  dom cdm 5591  ran crn 5592  (class class class)co 7295   ExId cexid 36030  Magmacmagm 36034

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 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7608

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-fo 6453  df-fv 6455  df-ov 7298  df-exid 36031  df-mgmOLD 36035

This theorem is referenced by:  exidu1  36042