Theorem opidonOLD 39052

Description: Obsolete version of mgmidpfod 18907 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)⟶onto𝑋)

Proof of Theorem opidonOLD

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

Step	Hyp	Ref	Expression
1		inss1 4136	. . . 4 ⊢ (Magma ∩ ExId ) ⊆ Magma
2	1	sseli 3909	. . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
3		opidonOLD.1	. . . . 5 ⊢ 𝑋 = dom dom 𝐺
4	3	ismgmOLD 39050	. . . 4 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
5	4	ibi 270	. . 3 ⊢ (𝐺 ∈ Magma → 𝐺:(𝑋 × 𝑋)⟶𝑋)
6	2, 5	syl 18	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7		inss2 4137	. . . . 5 ⊢ (Magma ∩ ExId ) ⊆ ExId
8	7	sseli 3909	. . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ ExId )
9	3	isexid 39047	. . . . 5 ⊢ (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
10	9	biimpd 232	. . . 4 ⊢ (𝐺 ∈ ExId → (𝐺 ∈ ExId → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
11	8, 8, 10	sylc 66	. . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
12		simpl 488	. . . . . . . 8 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
13	12	ralimi 3103	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
14		oveq2 7459	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
15		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
16	14, 15	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
17	16	rspcv 3576	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
18		eqcom 2770	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑥) = 𝑦)
19	14	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑦 ↔ (𝑢𝐺𝑦) = 𝑦))
20	18, 19	bitrid 286	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑦) = 𝑦))
21	20	rspcev 3580	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑋 ∧ (𝑢𝐺𝑦) = 𝑦) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
22	21	ex 418	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝑢𝐺𝑦) = 𝑦 → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
23	17, 22	syld 48	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
24	13, 23	syl5 35	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
25	24	reximdv 3181	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
26	25	impcom 413	. . . 4 ⊢ ((∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
27	26	ralrimiva 3158	. . 3 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
28	11, 27	syl 18	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
29		foov 7627	. 2 ⊢ (𝐺:(𝑋 × 𝑋)⟶onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
30	6, 28, 29	sylanbrc 595	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶onto𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ∩ cin 4092   × cxp 5656  dom cdm 5658  ⟶wf 6562  ⟶ontowfo 6564  (class class class)co 7451   ExId cexid 39044  Magmacmagm 39048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-ov 7454  df-exid 39045  df-mgmOLD 39049

This theorem is used by:  rngopidOLD  39053  opidon2OLD  39054