Theorem exidreslem 35729

Description: Lemma for exidres 35730 and exidresid 35731. (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 5762	. . . . . . 7 ⊢ dom 𝐻 = dom (𝐺 ↾ (𝑌 × 𝑌))
3		xpss12 5555	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
4	3	anidms 570	. . . . . . . . . 10 ⊢ (𝑌 ⊆ 𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
5		exidres.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ran 𝐺
6	5	opidon2OLD 35706	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
7		fof 6622	. . . . . . . . . . . 12 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → 𝐺:(𝑋 × 𝑋)⟶𝑋)
8		fdm 6543	. . . . . . . . . . . 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 3923	. . . . . . . . . 10 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)))
11	4, 10	syl5ibr 249	. . . . . . . . 9 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝑌 ⊆ 𝑋 → (𝑌 × 𝑌) ⊆ dom 𝐺))
12	11	imp 410	. . . . . . . 8 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ dom 𝐺)
13		ssdmres 5863	. . . . . . . 8 ⊢ ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
14	12, 13	sylib 221	. . . . . . 7 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
15	2, 14	syl5eq 2786	. . . . . 6 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → dom 𝐻 = (𝑌 × 𝑌))
16	15	dmeqd 5763	. . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → dom dom 𝐻 = dom (𝑌 × 𝑌))
17		dmxpid 5788	. . . . 5 ⊢ dom (𝑌 × 𝑌) = 𝑌
18	16, 17	eqtrdi 2790	. . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) → dom dom 𝐻 = 𝑌)
19	18	eleq2d 2819	. . 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 3886	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
22		exidres.2	. . . . . . . . . . 11 ⊢ 𝑈 = (GId‘𝐺)
23	5, 22	cmpidelt 35711	. . . . . . . . . 10 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑥 ∈ 𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
24	21, 23	sylan2 596	. . . . . . . . 9 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
25	24	anassrs 471	. . . . . . . 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 7215	. . . . . . . . . . 11 ⊢ (𝑈𝐻𝑥) = (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥)
28		ovres 7363	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥) = (𝑈𝐺𝑥))
29	27, 28	syl5eq 2786	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → (𝑈𝐻𝑥) = (𝑈𝐺𝑥))
30	29	eqeq1d 2736	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → ((𝑈𝐻𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
31	1	oveqi 7215	. . . . . . . . . . . 12 ⊢ (𝑥𝐻𝑈) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈)
32		ovres 7363	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑈 ∈ 𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈) = (𝑥𝐺𝑈))
33	31, 32	syl5eq 2786	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑈 ∈ 𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
34	33	ancoms 462	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
35	34	eqeq1d 2736	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → ((𝑥𝐻𝑈) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
36	30, 35	anbi12d 634	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
37	36	adantl 485	. . . . . . 7 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌)) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
38	26, 37	mpbird 260	. . . . . 6 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑈 ∈ 𝑌 ∧ 𝑥 ∈ 𝑌)) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
39	38	anassrs 471	. . . . 5 ⊢ ((((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑈 ∈ 𝑌) ∧ 𝑥 ∈ 𝑌) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
40	39	ralrimiva 3098	. . . 4 ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑈 ∈ 𝑌) → ∀𝑥 ∈ 𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
41	40	3impa 1112	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → ∀𝑥 ∈ 𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
42	12	3adant3 1134	. . . . . . . 8 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑌 × 𝑌) ⊆ dom 𝐺)
43	42, 13	sylib 221	. . . . . . 7 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
44	2, 43	syl5eq 2786	. . . . . 6 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → dom 𝐻 = (𝑌 × 𝑌))
45	44	dmeqd 5763	. . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → dom dom 𝐻 = dom (𝑌 × 𝑌))
46	45, 17	eqtrdi 2790	. . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → dom dom 𝐻 = 𝑌)
47	46	raleqdv 3318	. . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ∀𝑥 ∈ 𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
48	41, 47	mpbird 260	. 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
49	20, 48	jca 515	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 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3054   ∩ cin 3856   ⊆ wss 3857   × cxp 5538  dom cdm 5540  ran crn 5541   ↾ cres 5542  ⟶wf 6365  –onto→wfo 6367  ‘cfv 6369  (class class class)co 7202  GIdcgi 28543   ExId cexid 35696  Magmacmagm 35700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-fo 6375  df-fv 6377  df-riota 7159  df-ov 7205  df-gid 28547  df-exid 35697  df-mgmOLD 35701

This theorem is referenced by:  exidres  35730  exidresid  35731