Theorem exidreslem 37901

Description: Lemma for exidres 37902 and exidresid 37903. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)

Hypotheses

Ref	Expression
exidres.1	⊢ 𝑋 = ran 𝐺
exidres.2	⊢ 𝑈 = (GId‘𝐺)
exidres.3	⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))

Assertion

Ref	Expression
exidreslem	⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))

Distinct variable groups:   𝑥,𝐺   𝑥,𝑌   𝑥,𝑋   𝑥,𝑈   𝑥,𝐻

Proof of Theorem exidreslem

Step	Hyp	Ref	Expression
1		exidres.3	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))
2	1	dmeqi 5884	. . . . . . 7 ⊢ dom 𝐻 = dom (𝐺 ↾ (𝑌 × 𝑌))
3		xpss12 5669	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
4	3	anidms 566	. . . . . . . . . 10 ⊢ (𝑌 ⊆ 𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
5		exidres.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ran 𝐺
6	5	opidon2OLD 37878	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
7		fof 6790	. . . . . . . . . . . 12 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → 𝐺:(𝑋 × 𝑋)⟶𝑋)
8		fdm 6715	. . . . . . . . . . . 12 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
9	6, 7, 8	3syl 18	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → dom 𝐺 = (𝑋 × 𝑋))
10	9	sseq2d 3991	. . . . . . . . . 10 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)))
11	4, 10	imbitrrid 246	. . . . . . . . 9 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝑌 ⊆ 𝑋 → (𝑌 × 𝑌) ⊆ dom 𝐺))
12	11	imp 406	. . . . . . . 8 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ dom 𝐺)
13		ssdmres 6000	. . . . . . . 8 ⊢ ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
14	12, 13	sylib 218	. . . . . . 7 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
15	2, 14	eqtrid 2782	. . . . . 6 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → dom 𝐻 = (𝑌 × 𝑌))
16	15	dmeqd 5885	. . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → dom dom 𝐻 = dom (𝑌 × 𝑌))
17		dmxpid 5910	. . . . 5 ⊢ dom (𝑌 × 𝑌) = 𝑌
18	16, 17	eqtrdi 2786	. . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → dom dom 𝐻 = 𝑌)
19	18	eleq2d 2820	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → (𝑈 ∈ dom dom 𝐻 ↔ 𝑈 ∈ 𝑌))
20	19	biimp3ar 1472	. 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ dom dom 𝐻)
21		ssel2 3953	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
22		exidres.2	. . . . . . . . . . 11 ⊢ 𝑈 = (GId‘𝐺)
23	5, 22	cmpidelt 37883	. . . . . . . . . 10 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑥 ∈ 𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
24	21, 23	sylan2 593	. . . . . . . . 9 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
25	24	anassrs 467	. . . . . . . 8 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑌) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
26	25	adantrl 716	. . . . . . 7 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
27	1	oveqi 7418	. . . . . . . . . . 11 ⊢ (𝑈𝐻𝑥) = (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥)
28		ovres 7573	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥) = (𝑈𝐺𝑥))
29	27, 28	eqtrid 2782	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → (𝑈𝐻𝑥) = (𝑈𝐺𝑥))
30	29	eqeq1d 2737	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → ((𝑈𝐻𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
31	1	oveqi 7418	. . . . . . . . . . . 12 ⊢ (𝑥𝐻𝑈) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈)
32		ovres 7573	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑈 ∈ 𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈) = (𝑥𝐺𝑈))
33	31, 32	eqtrid 2782	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑈 ∈ 𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
34	33	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
35	34	eqeq1d 2737	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → ((𝑥𝐻𝑈) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
36	30, 35	anbi12d 632	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
37	36	adantl 481	. . . . . . 7 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌)) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
38	26, 37	mpbird 257	. . . . . 6 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌)) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
39	38	anassrs 467	. . . . 5 ⊢ ((((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑈 ∈ 𝑌) ∧ 𝑥 ∈ 𝑌) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
40	39	ralrimiva 3132	. . . 4 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑈 ∈ 𝑌) → ∀𝑥 ∈ 𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
41	40	3impa 1109	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → ∀𝑥 ∈ 𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
42	12	3adant3 1132	. . . . . . 7 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑌 × 𝑌) ⊆ dom 𝐺)
43	42, 13	sylib 218	. . . . . 6 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
44	2, 43	eqtrid 2782	. . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → dom 𝐻 = (𝑌 × 𝑌))
45	44	dmeqd 5885	. . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → dom dom 𝐻 = dom (𝑌 × 𝑌))
46	45, 17	eqtrdi 2786	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → dom dom 𝐻 = 𝑌)
47	41, 46	raleqtrrdv 3309	. 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
48	20, 47	jca 511	1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ∩ cin 3925   ⊆ wss 3926   × cxp 5652  dom cdm 5654  ran crn 5655   ↾ cres 5656  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7405  GIdcgi 30471   ExId cexid 37868  Magmacmagm 37872

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-riota 7362  df-ov 7408  df-gid 30475  df-exid 37869  df-mgmOLD 37873

This theorem is referenced by:  exidres  37902  exidresid  37903