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Theorem cycpmconjvlem 30843
 Description: Lemma for cycpmconjv 30844. (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 6593 . . . 4 (𝐹:𝐷1-1-onto𝐷 → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 funrel 6342 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
5 dfrel2 6014 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 221 . . . . . 6 (Fun 𝐹𝐹 = 𝐹)
76reseq1d 5818 . . . . 5 (Fun 𝐹 → (𝐹 ↾ (𝐷𝐵)) = (𝐹 ↾ (𝐷𝐵)))
87cnveqd 5711 . . . 4 (Fun 𝐹(𝐹 ↾ (𝐷𝐵)) = (𝐹 ↾ (𝐷𝐵)))
98coeq2d 5698 . . 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 4060 . . . . . 6 (𝜑 → (𝐷𝐵) ⊆ 𝐷)
12 f1odm 6595 . . . . . . 7 (𝐹:𝐷1-1-onto𝐷 → dom 𝐹 = 𝐷)
131, 12syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐷)
1411, 13sseqtrrd 3956 . . . . 5 (𝜑 → (𝐷𝐵) ⊆ dom 𝐹)
15 ssdmres 5842 . . . . 5 ((𝐷𝐵) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐷𝐵)) = (𝐷𝐵))
1614, 15sylib 221 . . . 4 (𝜑 → dom (𝐹 ↾ (𝐷𝐵)) = (𝐷𝐵))
17 ssidd 3938 . . . 4 (𝜑 → (𝐷𝐵) ⊆ (𝐷𝐵))
1816, 17eqsstrd 3953 . . 3 (𝜑 → dom (𝐹 ↾ (𝐷𝐵)) ⊆ (𝐷𝐵))
19 cores2 6080 . . 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 6603 . . . . . 6 (𝐹:𝐷1-1-onto𝐷𝐹:𝐷1-1-onto𝐷)
22 f1ofun 6593 . . . . . 6 (𝐹:𝐷1-1-onto𝐷 → Fun 𝐹)
231, 21, 223syl 18 . . . . 5 (𝜑 → Fun 𝐹)
24 ssidd 3938 . . . . . . . 8 (𝜑𝐷𝐷)
2524, 13sseqtrrd 3956 . . . . . . 7 (𝜑𝐷 ⊆ dom 𝐹)
26 fores 6576 . . . . . . 7 ((Fun 𝐹𝐷 ⊆ dom 𝐹) → (𝐹𝐷):𝐷onto→(𝐹𝐷))
273, 25, 26syl2anc 587 . . . . . 6 (𝜑 → (𝐹𝐷):𝐷onto→(𝐹𝐷))
28 df-ima 5533 . . . . . . 7 (𝐹𝐷) = ran (𝐹𝐷)
29 foeq3 6564 . . . . . . 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 3956 . . . . . . 7 (𝜑𝐵 ⊆ dom 𝐹)
34 fores 6576 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵):𝐵onto→(𝐹𝐵))
353, 33, 34syl2anc 587 . . . . . 6 (𝜑 → (𝐹𝐵):𝐵onto→(𝐹𝐵))
36 df-ima 5533 . . . . . . 7 (𝐹𝐵) = ran (𝐹𝐵)
37 foeq3 6564 . . . . . . 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 6611 . . . . 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 1368 . . . 4 (𝜑 → (𝐹 ↾ (𝐷𝐵)):(𝐷𝐵)–1-1-onto→(ran (𝐹𝐷) ∖ ran (𝐹𝐵)))
42 f1ofn 6592 . . . . . . . . 9 (𝐹:𝐷1-1-onto𝐷𝐹 Fn 𝐷)
43 fnresdm 6439 . . . . . . . . 9 (𝐹 Fn 𝐷 → (𝐹𝐷) = 𝐹)
441, 42, 433syl 18 . . . . . . . 8 (𝜑 → (𝐹𝐷) = 𝐹)
4544rneqd 5773 . . . . . . 7 (𝜑 → ran (𝐹𝐷) = ran 𝐹)
46 f1ofo 6598 . . . . . . . 8 (𝐹:𝐷1-1-onto𝐷𝐹:𝐷onto𝐷)
47 forn 6569 . . . . . . . 8 (𝐹:𝐷onto𝐷 → ran 𝐹 = 𝐷)
481, 46, 473syl 18 . . . . . . 7 (𝜑 → ran 𝐹 = 𝐷)
4945, 48eqtrd 2833 . . . . . 6 (𝜑 → ran (𝐹𝐷) = 𝐷)
5049difeq1d 4049 . . . . 5 (𝜑 → (ran (𝐹𝐷) ∖ ran (𝐹𝐵)) = (𝐷 ∖ ran (𝐹𝐵)))
5150f1oeq3d 6588 . . . 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 6617 . . 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 2841 1 (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∖ cdif 3878   ⊆ wss 3881   I cid 5425  ◡ccnv 5519  dom cdm 5520  ran crn 5521   ↾ cres 5522   “ cima 5523   ∘ ccom 5524  Rel wrel 5525  Fun wfun 6319   Fn wfn 6320  –onto→wfo 6323  –1-1-onto→wf1o 6324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332 This theorem is referenced by:  cycpmconjv  30844
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