Theorem opidonOLD 35129

Description: Obsolete version of mndpfo 17933 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 4204	. . . 4 ⊢ (Magma ∩ ExId ) ⊆ Magma
2	1	sseli 3962	. . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
3		opidonOLD.1	. . . . 5 ⊢ 𝑋 = dom dom 𝐺
4	3	ismgmOLD 35127	. . . 4 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
5	4	ibi 269	. . 3 ⊢ (𝐺 ∈ Magma → 𝐺:(𝑋 × 𝑋)⟶𝑋)
6	2, 5	syl 17	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7		inss2 4205	. . . . 5 ⊢ (Magma ∩ ExId ) ⊆ ExId
8	7	sseli 3962	. . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ ExId )
9	3	isexid 35124	. . . . 5 ⊢ (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
10	9	biimpd 231	. . . 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 485	. . . . . . . 8 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
13	12	ralimi 3160	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
14		oveq2 7163	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
15		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
16	14, 15	eqeq12d 2837	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
17	16	rspcv 3617	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
18		eqcom 2828	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑥) = 𝑦)
19	14	eqeq1d 2823	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑦 ↔ (𝑢𝐺𝑦) = 𝑦))
20	18, 19	syl5bb 285	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑦) = 𝑦))
21	20	rspcev 3622	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑋 ∧ (𝑢𝐺𝑦) = 𝑦) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
22	21	ex 415	. . . . . . . 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 3273	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
26	25	impcom 410	. . . 4 ⊢ ((∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
27	26	ralrimiva 3182	. . 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 7321	. 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
30	6, 28, 29	sylanbrc 585	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ∩ cin 3934   × cxp 5552  dom cdm 5554  ⟶wf 6350  –onto→wfo 6352  (class class class)co 7155   ExId cexid 35121  Magmacmagm 35125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-ov 7158  df-exid 35122  df-mgmOLD 35126

This theorem is referenced by:  rngopidOLD  35130  opidon2OLD  35131