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Theorem cycpmconjvlem 33161
Description: Lemma for cycpmconjv 33162. (Contributed by Thierry Arnoux, 9-Oct-2023.)
Hypotheses
Ref Expression
cycpmconjvlem.f (𝜑𝐹:𝐷1-1-onto𝐷)
cycpmconjvlem.b (𝜑𝐵𝐷)
Assertion
Ref Expression
cycpmconjvlem (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))

Proof of Theorem cycpmconjvlem
StepHypRef Expression
1 cycpmconjvlem.f . . . 4 (𝜑𝐹:𝐷1-1-onto𝐷)
2 f1ofun 6850 . . . 4 (𝐹:𝐷1-1-onto𝐷 → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 funrel 6583 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
5 dfrel2 6209 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 218 . . . . . 6 (Fun 𝐹𝐹 = 𝐹)
76reseq1d 5996 . . . . 5 (Fun 𝐹 → (𝐹 ↾ (𝐷𝐵)) = (𝐹 ↾ (𝐷𝐵)))
87cnveqd 5886 . . . 4 (Fun 𝐹(𝐹 ↾ (𝐷𝐵)) = (𝐹 ↾ (𝐷𝐵)))
98coeq2d 5873 . . 3 (Fun 𝐹 → ((𝐹 ↾ (𝐷𝐵)) ∘ (𝐹 ↾ (𝐷𝐵))) = ((𝐹 ↾ (𝐷𝐵)) ∘ (𝐹 ↾ (𝐷𝐵))))
103, 9syl 17 . 2 (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ (𝐹 ↾ (𝐷𝐵))) = ((𝐹 ↾ (𝐷𝐵)) ∘ (𝐹 ↾ (𝐷𝐵))))
11 difssd 4137 . . . . . 6 (𝜑 → (𝐷𝐵) ⊆ 𝐷)
12 f1odm 6852 . . . . . . 7 (𝐹:𝐷1-1-onto𝐷 → dom 𝐹 = 𝐷)
131, 12syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐷)
1411, 13sseqtrrd 4021 . . . . 5 (𝜑 → (𝐷𝐵) ⊆ dom 𝐹)
15 ssdmres 6031 . . . . 5 ((𝐷𝐵) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐷𝐵)) = (𝐷𝐵))
1614, 15sylib 218 . . . 4 (𝜑 → dom (𝐹 ↾ (𝐷𝐵)) = (𝐷𝐵))
17 ssidd 4007 . . . 4 (𝜑 → (𝐷𝐵) ⊆ (𝐷𝐵))
1816, 17eqsstrd 4018 . . 3 (𝜑 → dom (𝐹 ↾ (𝐷𝐵)) ⊆ (𝐷𝐵))
19 cores2 6279 . . 3 (dom (𝐹 ↾ (𝐷𝐵)) ⊆ (𝐷𝐵) → ((𝐹 ↾ (𝐷𝐵)) ∘ (𝐹 ↾ (𝐷𝐵))) = ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹))
2018, 19syl 17 . 2 (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ (𝐹 ↾ (𝐷𝐵))) = ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹))
21 f1ocnv 6860 . . . . . 6 (𝐹:𝐷1-1-onto𝐷𝐹:𝐷1-1-onto𝐷)
22 f1ofun 6850 . . . . . 6 (𝐹:𝐷1-1-onto𝐷 → Fun 𝐹)
231, 21, 223syl 18 . . . . 5 (𝜑 → Fun 𝐹)
24 ssidd 4007 . . . . . . . 8 (𝜑𝐷𝐷)
2524, 13sseqtrrd 4021 . . . . . . 7 (𝜑𝐷 ⊆ dom 𝐹)
26 fores 6830 . . . . . . 7 ((Fun 𝐹𝐷 ⊆ dom 𝐹) → (𝐹𝐷):𝐷onto→(𝐹𝐷))
273, 25, 26syl2anc 584 . . . . . 6 (𝜑 → (𝐹𝐷):𝐷onto→(𝐹𝐷))
28 df-ima 5698 . . . . . . 7 (𝐹𝐷) = ran (𝐹𝐷)
29 foeq3 6818 . . . . . . 7 ((𝐹𝐷) = ran (𝐹𝐷) → ((𝐹𝐷):𝐷onto→(𝐹𝐷) ↔ (𝐹𝐷):𝐷onto→ran (𝐹𝐷)))
3028, 29ax-mp 5 . . . . . 6 ((𝐹𝐷):𝐷onto→(𝐹𝐷) ↔ (𝐹𝐷):𝐷onto→ran (𝐹𝐷))
3127, 30sylib 218 . . . . 5 (𝜑 → (𝐹𝐷):𝐷onto→ran (𝐹𝐷))
32 cycpmconjvlem.b . . . . . . . 8 (𝜑𝐵𝐷)
3332, 13sseqtrrd 4021 . . . . . . 7 (𝜑𝐵 ⊆ dom 𝐹)
34 fores 6830 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵):𝐵onto→(𝐹𝐵))
353, 33, 34syl2anc 584 . . . . . 6 (𝜑 → (𝐹𝐵):𝐵onto→(𝐹𝐵))
36 df-ima 5698 . . . . . . 7 (𝐹𝐵) = ran (𝐹𝐵)
37 foeq3 6818 . . . . . . 7 ((𝐹𝐵) = ran (𝐹𝐵) → ((𝐹𝐵):𝐵onto→(𝐹𝐵) ↔ (𝐹𝐵):𝐵onto→ran (𝐹𝐵)))
3836, 37ax-mp 5 . . . . . 6 ((𝐹𝐵):𝐵onto→(𝐹𝐵) ↔ (𝐹𝐵):𝐵onto→ran (𝐹𝐵))
3935, 38sylib 218 . . . . 5 (𝜑 → (𝐹𝐵):𝐵onto→ran (𝐹𝐵))
40 resdif 6869 . . . . 5 ((Fun 𝐹 ∧ (𝐹𝐷):𝐷onto→ran (𝐹𝐷) ∧ (𝐹𝐵):𝐵onto→ran (𝐹𝐵)) → (𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(ran (𝐹𝐷) ∖ ran (𝐹𝐵)))
4123, 31, 39, 40syl3anc 1373 . . . 4 (𝜑 → (𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(ran (𝐹𝐷) ∖ ran (𝐹𝐵)))
42 f1ofn 6849 . . . . . . . . 9 (𝐹:𝐷1-1-onto𝐷𝐹 Fn 𝐷)
43 fnresdm 6687 . . . . . . . . 9 (𝐹 Fn 𝐷 → (𝐹𝐷) = 𝐹)
441, 42, 433syl 18 . . . . . . . 8 (𝜑 → (𝐹𝐷) = 𝐹)
4544rneqd 5949 . . . . . . 7 (𝜑 → ran (𝐹𝐷) = ran 𝐹)
46 f1ofo 6855 . . . . . . . 8 (𝐹:𝐷1-1-onto𝐷𝐹:𝐷onto𝐷)
47 forn 6823 . . . . . . . 8 (𝐹:𝐷onto𝐷 → ran 𝐹 = 𝐷)
481, 46, 473syl 18 . . . . . . 7 (𝜑 → ran 𝐹 = 𝐷)
4945, 48eqtrd 2777 . . . . . 6 (𝜑 → ran (𝐹𝐷) = 𝐷)
5049difeq1d 4125 . . . . 5 (𝜑 → (ran (𝐹𝐷) ∖ ran (𝐹𝐵)) = (𝐷 ∖ ran (𝐹𝐵)))
5150f1oeq3d 6845 . . . 4 (𝜑 → ((𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(ran (𝐹𝐷) ∖ ran (𝐹𝐵)) ↔ (𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹𝐵))))
5241, 51mpbid 232 . . 3 (𝜑 → (𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹𝐵)))
53 f1ococnv2 6875 . . 3 ((𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹𝐵)) → ((𝐹 ↾ (𝐷𝐵)) ∘ (𝐹 ↾ (𝐷𝐵))) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
5452, 53syl 17 . 2 (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ (𝐹 ↾ (𝐷𝐵))) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
5510, 20, 543eqtr3d 2785 1 (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  cdif 3948  wss 3951   I cid 5577  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  ccom 5689  Rel wrel 5690  Fun wfun 6555   Fn wfn 6556  ontowfo 6559  1-1-ontowf1o 6560
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-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by:  cycpmconjv  33162
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