Theorem cycpmconjvlem 31127

Description: Lemma for cycpmconjv 31128. (Contributed by Thierry Arnoux, 9-Oct-2023.)

Hypotheses

Ref	Expression
cycpmconjvlem.f	⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝐷)
cycpmconjvlem.b	⊢ (𝜑 → 𝐵 ⊆ 𝐷)

Assertion

Ref	Expression
cycpmconjvlem	⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵))))

Proof of Theorem cycpmconjvlem

Step	Hyp	Ref	Expression
1		cycpmconjvlem.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝐷)
2		f1ofun 6663	. . . 4 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → Fun 𝐹)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐹)
4		funrel 6397	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
5		dfrel2 6052	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
6	4, 5	sylib 221	. . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹)
7	6	reseq1d 5850	. . . . 5 ⊢ (Fun 𝐹 → (◡◡𝐹 ↾ (𝐷 ∖ 𝐵)) = (𝐹 ↾ (𝐷 ∖ 𝐵)))
8	7	cnveqd 5744	. . . 4 ⊢ (Fun 𝐹 → ◡(◡◡𝐹 ↾ (𝐷 ∖ 𝐵)) = ◡(𝐹 ↾ (𝐷 ∖ 𝐵)))
9	8	coeq2d 5731	. . 3 ⊢ (Fun 𝐹 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡(◡◡𝐹 ↾ (𝐷 ∖ 𝐵))) = ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡(𝐹 ↾ (𝐷 ∖ 𝐵))))
10	3, 9	syl 17	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡(◡◡𝐹 ↾ (𝐷 ∖ 𝐵))) = ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡(𝐹 ↾ (𝐷 ∖ 𝐵))))
11		difssd 4047	. . . . . 6 ⊢ (𝜑 → (𝐷 ∖ 𝐵) ⊆ 𝐷)
12		f1odm 6665	. . . . . . 7 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → dom 𝐹 = 𝐷)
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐷)
14	11, 13	sseqtrrd 3942	. . . . 5 ⊢ (𝜑 → (𝐷 ∖ 𝐵) ⊆ dom 𝐹)
15		ssdmres 5874	. . . . 5 ⊢ ((𝐷 ∖ 𝐵) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐷 ∖ 𝐵)) = (𝐷 ∖ 𝐵))
16	14, 15	sylib 221	. . . 4 ⊢ (𝜑 → dom (𝐹 ↾ (𝐷 ∖ 𝐵)) = (𝐷 ∖ 𝐵))
17		ssidd 3924	. . . 4 ⊢ (𝜑 → (𝐷 ∖ 𝐵) ⊆ (𝐷 ∖ 𝐵))
18	16, 17	eqsstrd 3939	. . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐷 ∖ 𝐵)) ⊆ (𝐷 ∖ 𝐵))
19		cores2 6123	. . 3 ⊢ (dom (𝐹 ↾ (𝐷 ∖ 𝐵)) ⊆ (𝐷 ∖ 𝐵) → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡(◡◡𝐹 ↾ (𝐷 ∖ 𝐵))) = ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹))
20	18, 19	syl 17	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡(◡◡𝐹 ↾ (𝐷 ∖ 𝐵))) = ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹))
21		f1ocnv 6673	. . . . . 6 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → ◡𝐹:𝐷–1-1-onto→𝐷)
22		f1ofun 6663	. . . . . 6 ⊢ (◡𝐹:𝐷–1-1-onto→𝐷 → Fun ◡𝐹)
23	1, 21, 22	3syl 18	. . . . 5 ⊢ (𝜑 → Fun ◡𝐹)
24		ssidd 3924	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ 𝐷)
25	24, 13	sseqtrrd 3942	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)
26		fores 6643	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐷 ⊆ dom 𝐹) → (𝐹 ↾ 𝐷):𝐷–onto→(𝐹 “ 𝐷))
27	3, 25, 26	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→(𝐹 “ 𝐷))
28		df-ima 5564	. . . . . . 7 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
29		foeq3 6631	. . . . . . 7 ⊢ ((𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷) → ((𝐹 ↾ 𝐷):𝐷–onto→(𝐹 “ 𝐷) ↔ (𝐹 ↾ 𝐷):𝐷–onto→ran (𝐹 ↾ 𝐷)))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ ((𝐹 ↾ 𝐷):𝐷–onto→(𝐹 “ 𝐷) ↔ (𝐹 ↾ 𝐷):𝐷–onto→ran (𝐹 ↾ 𝐷))
31	27, 30	sylib 221	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→ran (𝐹 ↾ 𝐷))
32		cycpmconjvlem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝐷)
33	32, 13	sseqtrrd 3942	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ dom 𝐹)
34		fores 6643	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 ↾ 𝐵):𝐵–onto→(𝐹 “ 𝐵))
35	3, 33, 34	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵–onto→(𝐹 “ 𝐵))
36		df-ima 5564	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
37		foeq3 6631	. . . . . . 7 ⊢ ((𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵):𝐵–onto→(𝐹 “ 𝐵) ↔ (𝐹 ↾ 𝐵):𝐵–onto→ran (𝐹 ↾ 𝐵)))
38	36, 37	ax-mp 5	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵):𝐵–onto→(𝐹 “ 𝐵) ↔ (𝐹 ↾ 𝐵):𝐵–onto→ran (𝐹 ↾ 𝐵))
39	35, 38	sylib 221	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵–onto→ran (𝐹 ↾ 𝐵))
40		resdif 6681	. . . . 5 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐷):𝐷–onto→ran (𝐹 ↾ 𝐷) ∧ (𝐹 ↾ 𝐵):𝐵–onto→ran (𝐹 ↾ 𝐵)) → (𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(ran (𝐹 ↾ 𝐷) ∖ ran (𝐹 ↾ 𝐵)))
41	23, 31, 39, 40	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(ran (𝐹 ↾ 𝐷) ∖ ran (𝐹 ↾ 𝐵)))
42		f1ofn 6662	. . . . . . . . 9 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹 Fn 𝐷)
43		fnresdm 6496	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐷 → (𝐹 ↾ 𝐷) = 𝐹)
44	1, 42, 43	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = 𝐹)
45	44	rneqd 5807	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = ran 𝐹)
46		f1ofo 6668	. . . . . . . 8 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷–onto→𝐷)
47		forn 6636	. . . . . . . 8 ⊢ (𝐹:𝐷–onto→𝐷 → ran 𝐹 = 𝐷)
48	1, 46, 47	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐷)
49	45, 48	eqtrd 2777	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐷)
50	49	difeq1d 4036	. . . . 5 ⊢ (𝜑 → (ran (𝐹 ↾ 𝐷) ∖ ran (𝐹 ↾ 𝐵)) = (𝐷 ∖ ran (𝐹 ↾ 𝐵)))
51	50	f1oeq3d 6658	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(ran (𝐹 ↾ 𝐷) ∖ ran (𝐹 ↾ 𝐵)) ↔ (𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹 ↾ 𝐵))))
52	41, 51	mpbid 235	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹 ↾ 𝐵)))
53		f1ococnv2 6687	. . 3 ⊢ ((𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹 ↾ 𝐵)) → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡(𝐹 ↾ (𝐷 ∖ 𝐵))) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵))))
54	52, 53	syl 17	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡(𝐹 ↾ (𝐷 ∖ 𝐵))) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵))))
55	10, 20, 54	3eqtr3d 2785	1 ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1543   ∖ cdif 3863   ⊆ wss 3866   I cid 5454  ^◡ccnv 5550  dom cdm 5551  ran crn 5552   ↾ cres 5553   “ cima 5554   ∘ ccom 5555  Rel wrel 5556  Fun wfun 6374   Fn wfn 6375  –onto→wfo 6378  –1-1-onto→wf1o 6379

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 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387

This theorem is referenced by:  cycpmconjv  31128