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Theorem cycpmconjvlem 31127
Description: Lemma for cycpmconjv 31128. (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 6663 . . . 4 (𝐹:𝐷1-1-onto𝐷 → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 funrel 6397 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
5 dfrel2 6052 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 221 . . . . . 6 (Fun 𝐹𝐹 = 𝐹)
76reseq1d 5850 . . . . 5 (Fun 𝐹 → (𝐹 ↾ (𝐷𝐵)) = (𝐹 ↾ (𝐷𝐵)))
87cnveqd 5744 . . . 4 (Fun 𝐹(𝐹 ↾ (𝐷𝐵)) = (𝐹 ↾ (𝐷𝐵)))
98coeq2d 5731 . . 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 4047 . . . . . 6 (𝜑 → (𝐷𝐵) ⊆ 𝐷)
12 f1odm 6665 . . . . . . 7 (𝐹:𝐷1-1-onto𝐷 → dom 𝐹 = 𝐷)
131, 12syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐷)
1411, 13sseqtrrd 3942 . . . . 5 (𝜑 → (𝐷𝐵) ⊆ dom 𝐹)
15 ssdmres 5874 . . . . 5 ((𝐷𝐵) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐷𝐵)) = (𝐷𝐵))
1614, 15sylib 221 . . . 4 (𝜑 → dom (𝐹 ↾ (𝐷𝐵)) = (𝐷𝐵))
17 ssidd 3924 . . . 4 (𝜑 → (𝐷𝐵) ⊆ (𝐷𝐵))
1816, 17eqsstrd 3939 . . 3 (𝜑 → dom (𝐹 ↾ (𝐷𝐵)) ⊆ (𝐷𝐵))
19 cores2 6123 . . 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 6673 . . . . . 6 (𝐹:𝐷1-1-onto𝐷𝐹:𝐷1-1-onto𝐷)
22 f1ofun 6663 . . . . . 6 (𝐹:𝐷1-1-onto𝐷 → Fun 𝐹)
231, 21, 223syl 18 . . . . 5 (𝜑 → Fun 𝐹)
24 ssidd 3924 . . . . . . . 8 (𝜑𝐷𝐷)
2524, 13sseqtrrd 3942 . . . . . . 7 (𝜑𝐷 ⊆ dom 𝐹)
26 fores 6643 . . . . . . 7 ((Fun 𝐹𝐷 ⊆ dom 𝐹) → (𝐹𝐷):𝐷onto→(𝐹𝐷))
273, 25, 26syl2anc 587 . . . . . 6 (𝜑 → (𝐹𝐷):𝐷onto→(𝐹𝐷))
28 df-ima 5564 . . . . . . 7 (𝐹𝐷) = ran (𝐹𝐷)
29 foeq3 6631 . . . . . . 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 221 . . . . 5 (𝜑 → (𝐹𝐷):𝐷onto→ran (𝐹𝐷))
32 cycpmconjvlem.b . . . . . . . 8 (𝜑𝐵𝐷)
3332, 13sseqtrrd 3942 . . . . . . 7 (𝜑𝐵 ⊆ dom 𝐹)
34 fores 6643 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵):𝐵onto→(𝐹𝐵))
353, 33, 34syl2anc 587 . . . . . 6 (𝜑 → (𝐹𝐵):𝐵onto→(𝐹𝐵))
36 df-ima 5564 . . . . . . 7 (𝐹𝐵) = ran (𝐹𝐵)
37 foeq3 6631 . . . . . . 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 221 . . . . 5 (𝜑 → (𝐹𝐵):𝐵onto→ran (𝐹𝐵))
40 resdif 6681 . . . . 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 6662 . . . . . . . . 9 (𝐹:𝐷1-1-onto𝐷𝐹 Fn 𝐷)
43 fnresdm 6496 . . . . . . . . 9 (𝐹 Fn 𝐷 → (𝐹𝐷) = 𝐹)
441, 42, 433syl 18 . . . . . . . 8 (𝜑 → (𝐹𝐷) = 𝐹)
4544rneqd 5807 . . . . . . 7 (𝜑 → ran (𝐹𝐷) = ran 𝐹)
46 f1ofo 6668 . . . . . . . 8 (𝐹:𝐷1-1-onto𝐷𝐹:𝐷onto𝐷)
47 forn 6636 . . . . . . . 8 (𝐹:𝐷onto𝐷 → ran 𝐹 = 𝐷)
481, 46, 473syl 18 . . . . . . 7 (𝜑 → ran 𝐹 = 𝐷)
4945, 48eqtrd 2777 . . . . . 6 (𝜑 → ran (𝐹𝐷) = 𝐷)
5049difeq1d 4036 . . . . 5 (𝜑 → (ran (𝐹𝐷) ∖ ran (𝐹𝐵)) = (𝐷 ∖ ran (𝐹𝐵)))
5150f1oeq3d 6658 . . . 4 (𝜑 → ((𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(ran (𝐹𝐷) ∖ ran (𝐹𝐵)) ↔ (𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹𝐵))))
5241, 51mpbid 235 . . 3 (𝜑 → (𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(𝐷 ∖ ran (𝐹𝐵)))
53 f1ococnv2 6687 . . 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 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  ontowfo 6378  1-1-ontowf1o 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
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