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Theorem cycpmconjvlem 33144
Description: Lemma for cycpmconjv 33145. (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 6851 . . . 4 (𝐹:𝐷1-1-onto𝐷 → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 funrel 6585 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
5 dfrel2 6211 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 218 . . . . . 6 (Fun 𝐹𝐹 = 𝐹)
76reseq1d 5999 . . . . 5 (Fun 𝐹 → (𝐹 ↾ (𝐷𝐵)) = (𝐹 ↾ (𝐷𝐵)))
87cnveqd 5889 . . . 4 (Fun 𝐹(𝐹 ↾ (𝐷𝐵)) = (𝐹 ↾ (𝐷𝐵)))
98coeq2d 5876 . . 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 4147 . . . . . 6 (𝜑 → (𝐷𝐵) ⊆ 𝐷)
12 f1odm 6853 . . . . . . 7 (𝐹:𝐷1-1-onto𝐷 → dom 𝐹 = 𝐷)
131, 12syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐷)
1411, 13sseqtrrd 4037 . . . . 5 (𝜑 → (𝐷𝐵) ⊆ dom 𝐹)
15 ssdmres 6033 . . . . 5 ((𝐷𝐵) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐷𝐵)) = (𝐷𝐵))
1614, 15sylib 218 . . . 4 (𝜑 → dom (𝐹 ↾ (𝐷𝐵)) = (𝐷𝐵))
17 ssidd 4019 . . . 4 (𝜑 → (𝐷𝐵) ⊆ (𝐷𝐵))
1816, 17eqsstrd 4034 . . 3 (𝜑 → dom (𝐹 ↾ (𝐷𝐵)) ⊆ (𝐷𝐵))
19 cores2 6281 . . 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 6861 . . . . . 6 (𝐹:𝐷1-1-onto𝐷𝐹:𝐷1-1-onto𝐷)
22 f1ofun 6851 . . . . . 6 (𝐹:𝐷1-1-onto𝐷 → Fun 𝐹)
231, 21, 223syl 18 . . . . 5 (𝜑 → Fun 𝐹)
24 ssidd 4019 . . . . . . . 8 (𝜑𝐷𝐷)
2524, 13sseqtrrd 4037 . . . . . . 7 (𝜑𝐷 ⊆ dom 𝐹)
26 fores 6831 . . . . . . 7 ((Fun 𝐹𝐷 ⊆ dom 𝐹) → (𝐹𝐷):𝐷onto→(𝐹𝐷))
273, 25, 26syl2anc 584 . . . . . 6 (𝜑 → (𝐹𝐷):𝐷onto→(𝐹𝐷))
28 df-ima 5702 . . . . . . 7 (𝐹𝐷) = ran (𝐹𝐷)
29 foeq3 6819 . . . . . . 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 4037 . . . . . . 7 (𝜑𝐵 ⊆ dom 𝐹)
34 fores 6831 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵):𝐵onto→(𝐹𝐵))
353, 33, 34syl2anc 584 . . . . . 6 (𝜑 → (𝐹𝐵):𝐵onto→(𝐹𝐵))
36 df-ima 5702 . . . . . . 7 (𝐹𝐵) = ran (𝐹𝐵)
37 foeq3 6819 . . . . . . 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 6870 . . . . 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 1370 . . . 4 (𝜑 → (𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(ran (𝐹𝐷) ∖ ran (𝐹𝐵)))
42 f1ofn 6850 . . . . . . . . 9 (𝐹:𝐷1-1-onto𝐷𝐹 Fn 𝐷)
43 fnresdm 6688 . . . . . . . . 9 (𝐹 Fn 𝐷 → (𝐹𝐷) = 𝐹)
441, 42, 433syl 18 . . . . . . . 8 (𝜑 → (𝐹𝐷) = 𝐹)
4544rneqd 5952 . . . . . . 7 (𝜑 → ran (𝐹𝐷) = ran 𝐹)
46 f1ofo 6856 . . . . . . . 8 (𝐹:𝐷1-1-onto𝐷𝐹:𝐷onto𝐷)
47 forn 6824 . . . . . . . 8 (𝐹:𝐷onto𝐷 → ran 𝐹 = 𝐷)
481, 46, 473syl 18 . . . . . . 7 (𝜑 → ran 𝐹 = 𝐷)
4945, 48eqtrd 2775 . . . . . 6 (𝜑 → ran (𝐹𝐷) = 𝐷)
5049difeq1d 4135 . . . . 5 (𝜑 → (ran (𝐹𝐷) ∖ ran (𝐹𝐵)) = (𝐷 ∖ ran (𝐹𝐵)))
5150f1oeq3d 6846 . . . 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 6876 . . 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 2783 1 (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  cdif 3960  wss 3963   I cid 5582  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  ccom 5693  Rel wrel 5694  Fun wfun 6557   Fn wfn 6558  ontowfo 6561  1-1-ontowf1o 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  cycpmconjv  33145
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