Theorem opidonOLD 37600

Description: Obsolete version of mndpfo 18771 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
opidonOLD.1	⊢ 𝑋 = dom dom 𝐺

Assertion

Ref	Expression
opidonOLD	⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)

Proof of Theorem opidonOLD

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4240	. . . 4 ⊢ (Magma ∩ ExId ) ⊆ Magma
2	1	sseli 3987	. . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
3		opidonOLD.1	. . . . 5 ⊢ 𝑋 = dom dom 𝐺
4	3	ismgmOLD 37598	. . . 4 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
5	4	ibi 266	. . 3 ⊢ (𝐺 ∈ Magma → 𝐺:(𝑋 × 𝑋)⟶𝑋)
6	2, 5	syl 17	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7		inss2 4241	. . . . 5 ⊢ (Magma ∩ ExId ) ⊆ ExId
8	7	sseli 3987	. . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ ExId )
9	3	isexid 37595	. . . . 5 ⊢ (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
10	9	biimpd 228	. . . 4 ⊢ (𝐺 ∈ ExId → (𝐺 ∈ ExId → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
11	8, 8, 10	sylc 65	. . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
12		simpl 481	. . . . . . . 8 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
13	12	ralimi 3076	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
14		oveq2 7438	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
15		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
16	14, 15	eqeq12d 2745	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
17	16	rspcv 3616	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
18		eqcom 2736	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑥) = 𝑦)
19	14	eqeq1d 2731	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑦 ↔ (𝑢𝐺𝑦) = 𝑦))
20	18, 19	bitrid 282	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑦) = 𝑦))
21	20	rspcev 3620	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑋 ∧ (𝑢𝐺𝑦) = 𝑦) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
22	21	ex 411	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝑢𝐺𝑦) = 𝑦 → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
23	17, 22	syld 47	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
24	13, 23	syl5 34	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
25	24	reximdv 3163	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
26	25	impcom 406	. . . 4 ⊢ ((∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
27	26	ralrimiva 3139	. . 3 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
28	11, 27	syl 17	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
29		foov 7605	. 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
30	6, 28, 29	sylanbrc 581	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2100  ∀wral 3054  ∃wrex 3063   ∩ cin 3958   × cxp 5684  dom cdm 5686  ⟶wf 6554  –onto→wfo 6556  (class class class)co 7430   ExId cexid 37592  Magmacmagm 37596

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 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pr 5437  ax-un 7751

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-nul 4336  df-if 4537  df-sn 4637  df-pr 4639  df-op 4643  df-uni 4919  df-iun 5008  df-br 5157  df-opab 5219  df-mpt 5240  df-id 5584  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-fo 6564  df-fv 6566  df-ov 7433  df-exid 37593  df-mgmOLD 37597

This theorem is referenced by:  rngopidOLD  37601  opidon2OLD  37602