Theorem cycpmconjvlem 33134

Description: Lemma for cycpmconjv 33135. (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 6864	. . . 4 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → Fun 𝐹)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐹)
4		funrel 6595	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
5		dfrel2 6220	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
6	4, 5	sylib 218	. . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹)
7	6	reseq1d 6008	. . . . 5 ⊢ (Fun 𝐹 → (◡◡𝐹 ↾ (𝐷 ∖ 𝐵)) = (𝐹 ↾ (𝐷 ∖ 𝐵)))
8	7	cnveqd 5900	. . . 4 ⊢ (Fun 𝐹 → ◡(◡◡𝐹 ↾ (𝐷 ∖ 𝐵)) = ◡(𝐹 ↾ (𝐷 ∖ 𝐵)))
9	8	coeq2d 5887	. . 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 4160	. . . . . 6 ⊢ (𝜑 → (𝐷 ∖ 𝐵) ⊆ 𝐷)
12		f1odm 6866	. . . . . . 7 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → dom 𝐹 = 𝐷)
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐷)
14	11, 13	sseqtrrd 4050	. . . . 5 ⊢ (𝜑 → (𝐷 ∖ 𝐵) ⊆ dom 𝐹)
15		ssdmres 6042	. . . . 5 ⊢ ((𝐷 ∖ 𝐵) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐷 ∖ 𝐵)) = (𝐷 ∖ 𝐵))
16	14, 15	sylib 218	. . . 4 ⊢ (𝜑 → dom (𝐹 ↾ (𝐷 ∖ 𝐵)) = (𝐷 ∖ 𝐵))
17		ssidd 4032	. . . 4 ⊢ (𝜑 → (𝐷 ∖ 𝐵) ⊆ (𝐷 ∖ 𝐵))
18	16, 17	eqsstrd 4047	. . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐷 ∖ 𝐵)) ⊆ (𝐷 ∖ 𝐵))
19		cores2 6290	. . 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 6874	. . . . . 6 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → ◡𝐹:𝐷–1-1-onto→𝐷)
22		f1ofun 6864	. . . . . 6 ⊢ (◡𝐹:𝐷–1-1-onto→𝐷 → Fun ◡𝐹)
23	1, 21, 22	3syl 18	. . . . 5 ⊢ (𝜑 → Fun ◡𝐹)
24		ssidd 4032	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ 𝐷)
25	24, 13	sseqtrrd 4050	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)
26		fores 6844	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐷 ⊆ dom 𝐹) → (𝐹 ↾ 𝐷):𝐷–onto→(𝐹 “ 𝐷))
27	3, 25, 26	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→(𝐹 “ 𝐷))
28		df-ima 5713	. . . . . . 7 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
29		foeq3 6832	. . . . . . 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 218	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→ran (𝐹 ↾ 𝐷))
32		cycpmconjvlem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝐷)
33	32, 13	sseqtrrd 4050	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ dom 𝐹)
34		fores 6844	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 ↾ 𝐵):𝐵–onto→(𝐹 “ 𝐵))
35	3, 33, 34	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵–onto→(𝐹 “ 𝐵))
36		df-ima 5713	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
37		foeq3 6832	. . . . . . 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 218	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵–onto→ran (𝐹 ↾ 𝐵))
40		resdif 6883	. . . . 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 1371	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(ran (𝐹 ↾ 𝐷) ∖ ran (𝐹 ↾ 𝐵)))
42		f1ofn 6863	. . . . . . . . 9 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹 Fn 𝐷)
43		fnresdm 6699	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐷 → (𝐹 ↾ 𝐷) = 𝐹)
44	1, 42, 43	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = 𝐹)
45	44	rneqd 5963	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = ran 𝐹)
46		f1ofo 6869	. . . . . . . 8 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷–onto→𝐷)
47		forn 6837	. . . . . . . 8 ⊢ (𝐹:𝐷–onto→𝐷 → ran 𝐹 = 𝐷)
48	1, 46, 47	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐷)
49	45, 48	eqtrd 2780	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐷)
50	49	difeq1d 4148	. . . . 5 ⊢ (𝜑 → (ran (𝐹 ↾ 𝐷) ∖ ran (𝐹 ↾ 𝐵)) = (𝐷 ∖ ran (𝐹 ↾ 𝐵)))
51	50	f1oeq3d 6859	. . . 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 232	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹 ↾ 𝐵)))
53		f1ococnv2 6889	. . 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 2788	1 ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∖ cdif 3973   ⊆ wss 3976   I cid 5592  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703   ∘ ccom 5704  Rel wrel 5705  Fun wfun 6567   Fn wfn 6568  –onto→wfo 6571  –1-1-onto→wf1o 6572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580

This theorem is referenced by:  cycpmconjv  33135