Theorem cycpmconjvlem 33229

Description: Lemma for cycpmconjv 33230. (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 6776	. . . 4 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → Fun 𝐹)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐹)
4		funrel 6509	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
5		dfrel2 6147	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
6	4, 5	sylib 219	. . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹)
7	6	reseq1d 5937	. . . . 5 ⊢ (Fun 𝐹 → (◡◡𝐹 ↾ (𝐷 ∖ 𝐵)) = (𝐹 ↾ (𝐷 ∖ 𝐵)))
8	7	cnveqd 5824	. . . 4 ⊢ (Fun 𝐹 → ◡(◡◡𝐹 ↾ (𝐷 ∖ 𝐵)) = ◡(𝐹 ↾ (𝐷 ∖ 𝐵)))
9	8	coeq2d 5811	. . 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 4074	. . . . . 6 ⊢ (𝜑 → (𝐷 ∖ 𝐵) ⊆ 𝐷)
12		f1odm 6778	. . . . . . 7 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → dom 𝐹 = 𝐷)
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐷)
14	11, 13	sseqtrrd 3959	. . . . 5 ⊢ (𝜑 → (𝐷 ∖ 𝐵) ⊆ dom 𝐹)
15		ssdmres 5972	. . . . 5 ⊢ ((𝐷 ∖ 𝐵) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐷 ∖ 𝐵)) = (𝐷 ∖ 𝐵))
16	14, 15	sylib 219	. . . 4 ⊢ (𝜑 → dom (𝐹 ↾ (𝐷 ∖ 𝐵)) = (𝐷 ∖ 𝐵))
17		ssidd 3945	. . . 4 ⊢ (𝜑 → (𝐷 ∖ 𝐵) ⊆ (𝐷 ∖ 𝐵))
18	16, 17	eqsstrd 3956	. . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐷 ∖ 𝐵)) ⊆ (𝐷 ∖ 𝐵))
19		cores2 6218	. . 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 6786	. . . . . 6 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → ◡𝐹:𝐷–1-1-onto→𝐷)
22		f1ofun 6776	. . . . . 6 ⊢ (◡𝐹:𝐷–1-1-onto→𝐷 → Fun ◡𝐹)
23	1, 21, 22	3syl 18	. . . . 5 ⊢ (𝜑 → Fun ◡𝐹)
24		ssidd 3945	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ 𝐷)
25	24, 13	sseqtrrd 3959	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)
26		fores 6756	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐷 ⊆ dom 𝐹) → (𝐹 ↾ 𝐷):𝐷–onto→(𝐹 “ 𝐷))
27	3, 25, 26	syl2anc 590	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→(𝐹 “ 𝐷))
28		df-ima 5638	. . . . . . 7 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
29		foeq3 6744	. . . . . . 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 219	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→ran (𝐹 ↾ 𝐷))
32		cycpmconjvlem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝐷)
33	32, 13	sseqtrrd 3959	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ dom 𝐹)
34		fores 6756	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 ↾ 𝐵):𝐵–onto→(𝐹 “ 𝐵))
35	3, 33, 34	syl2anc 590	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵–onto→(𝐹 “ 𝐵))
36		df-ima 5638	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
37		foeq3 6744	. . . . . . 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 219	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵–onto→ran (𝐹 ↾ 𝐵))
40		resdif 6795	. . . . 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 1379	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(ran (𝐹 ↾ 𝐷) ∖ ran (𝐹 ↾ 𝐵)))
42		f1ofn 6775	. . . . . . . . 9 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹 Fn 𝐷)
43		fnresdm 6611	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐷 → (𝐹 ↾ 𝐷) = 𝐹)
44	1, 42, 43	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = 𝐹)
45	44	rneqd 5887	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = ran 𝐹)
46		f1ofo 6781	. . . . . . . 8 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷–onto→𝐷)
47		forn 6749	. . . . . . . 8 ⊢ (𝐹:𝐷–onto→𝐷 → ran 𝐹 = 𝐷)
48	1, 46, 47	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐷)
49	45, 48	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐷)
50	49	difeq1d 4063	. . . . 5 ⊢ (𝜑 → (ran (𝐹 ↾ 𝐷) ∖ ran (𝐹 ↾ 𝐵)) = (𝐷 ∖ ran (𝐹 ↾ 𝐵)))
51	50	f1oeq3d 6771	. . . 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 233	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐷 ∖ 𝐵)):(𝐷 ∖ 𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹 ↾ 𝐵)))
53		f1ococnv2 6801	. . 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 2783	1 ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∖ cdif 3887   ⊆ wss 3890   I cid 5519  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Rel wrel 5630  Fun wfun 6486   Fn wfn 6487  –onto→wfo 6490  –1-1-onto→wf1o 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499

This theorem is referenced by:  cycpmconjv  33230