Theorem isexid2 37263

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 37261	. . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
3		elin 3960	. . . . . . 7 ⊢ (𝐺 ∈ (Magma ∩ ExId ) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ ExId ))
4		eqid 2727	. . . . . . . . . . 11 ⊢ dom dom 𝐺 = dom dom 𝐺
5	4	isexid 37255	. . . . . . . . . 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 216	. . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
10		eqeq2 2739	. . . . . . 7 ⊢ (ran 𝐺 = dom dom 𝐺 → (𝑋 = ran 𝐺 ↔ 𝑋 = dom dom 𝐺))
11		raleq 3317	. . . . . . . 8 ⊢ (ran 𝐺 = dom dom 𝐺 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
12	11	rexeqbi1dv 3329	. . . . . . 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 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 3317	. . . 4 ⊢ (𝑋 = ran 𝐺 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
18	17	rexeqbi1dv 3329	. . 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 1534   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065   ∩ cin 3943  dom cdm 5672  ran crn 5673  (class class class)co 7414   ExId cexid 37252  Magmacmagm 37256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-ov 7417  df-exid 37253  df-mgmOLD 37257

This theorem is referenced by:  exidu1  37264