Theorem opidonOLD 37853

Description: Obsolete version of mndpfo 18691 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 4203	. . . 4 ⊢ (Magma ∩ ExId ) ⊆ Magma
2	1	sseli 3945	. . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
3		opidonOLD.1	. . . . 5 ⊢ 𝑋 = dom dom 𝐺
4	3	ismgmOLD 37851	. . . 4 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
5	4	ibi 267	. . 3 ⊢ (𝐺 ∈ Magma → 𝐺:(𝑋 × 𝑋)⟶𝑋)
6	2, 5	syl 17	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7		inss2 4204	. . . . 5 ⊢ (Magma ∩ ExId ) ⊆ ExId
8	7	sseli 3945	. . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ ExId )
9	3	isexid 37848	. . . . 5 ⊢ (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
10	9	biimpd 229	. . . 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 482	. . . . . . . 8 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
13	12	ralimi 3067	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
14		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
15		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
16	14, 15	eqeq12d 2746	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
17	16	rspcv 3587	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
18		eqcom 2737	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑥) = 𝑦)
19	14	eqeq1d 2732	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑦 ↔ (𝑢𝐺𝑦) = 𝑦))
20	18, 19	bitrid 283	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑦) = 𝑦))
21	20	rspcev 3591	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑋 ∧ (𝑢𝐺𝑦) = 𝑦) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
22	21	ex 412	. . . . . . . 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 3149	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
26	25	impcom 407	. . . 4 ⊢ ((∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥))
27	26	ralrimiva 3126	. . 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 7566	. 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑥 ∈ 𝑋 𝑦 = (𝑢𝐺𝑥)))
30	6, 28, 29	sylanbrc 583	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ∩ cin 3916   × cxp 5639  dom cdm 5641  ⟶wf 6510  –onto→wfo 6512  (class class class)co 7390   ExId cexid 37845  Magmacmagm 37849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-ov 7393  df-exid 37846  df-mgmOLD 37850

This theorem is referenced by:  rngopidOLD  37854  opidon2OLD  37855