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Theorem yonedalem22 18228
Description: Lemma for yoneda 18233. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y π‘Œ = (Yonβ€˜πΆ)
yoneda.b 𝐡 = (Baseβ€˜πΆ)
yoneda.1 1 = (Idβ€˜πΆ)
yoneda.o 𝑂 = (oppCatβ€˜πΆ)
yoneda.s 𝑆 = (SetCatβ€˜π‘ˆ)
yoneda.t 𝑇 = (SetCatβ€˜π‘‰)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomFβ€˜π‘„)
yoneda.r 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (πœ‘ β†’ 𝐢 ∈ Cat)
yoneda.w (πœ‘ β†’ 𝑉 ∈ π‘Š)
yoneda.u (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
yoneda.v (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
yonedalem21.f (πœ‘ β†’ 𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
yonedalem22.g (πœ‘ β†’ 𝐺 ∈ (𝑂 Func 𝑆))
yonedalem22.p (πœ‘ β†’ 𝑃 ∈ 𝐡)
yonedalem22.a (πœ‘ β†’ 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
yonedalem22.k (πœ‘ β†’ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋))
Assertion
Ref Expression
yonedalem22 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾) = (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(2nd β€˜π»)⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)𝐴))

Proof of Theorem yonedalem22
StepHypRef Expression
1 yoneda.z . . . . . . 7 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
21fveq2i 6892 . . . . . 6 (2nd β€˜π‘) = (2nd β€˜(𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
32oveqi 7419 . . . . 5 (⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©) = (⟨𝐹, π‘‹βŸ©(2nd β€˜(𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, π‘ƒβŸ©)
43oveqi 7419 . . . 4 (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾) = (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜(𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, π‘ƒβŸ©)𝐾)
5 df-ov 7409 . . . 4 (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜(𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, π‘ƒβŸ©)𝐾) = ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)
64, 5eqtri 2761 . . 3 (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾) = ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)
7 eqid 2733 . . . . 5 (𝑄 Γ—c 𝑂) = (𝑄 Γ—c 𝑂)
8 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
98fucbas 17909 . . . . 5 (𝑂 Func 𝑆) = (Baseβ€˜π‘„)
10 yoneda.o . . . . . 6 𝑂 = (oppCatβ€˜πΆ)
11 yoneda.b . . . . . 6 𝐡 = (Baseβ€˜πΆ)
1210, 11oppcbas 17660 . . . . 5 𝐡 = (Baseβ€˜π‘‚)
137, 9, 12xpcbas 18127 . . . 4 ((𝑂 Func 𝑆) Γ— 𝐡) = (Baseβ€˜(𝑄 Γ—c 𝑂))
14 eqid 2733 . . . . 5 ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) = ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))
15 eqid 2733 . . . . 5 ((oppCatβ€˜π‘„) Γ—c 𝑄) = ((oppCatβ€˜π‘„) Γ—c 𝑄)
16 yoneda.c . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ Cat)
1710oppccat 17665 . . . . . . . . 9 (𝐢 ∈ Cat β†’ 𝑂 ∈ Cat)
1816, 17syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑂 ∈ Cat)
19 yoneda.w . . . . . . . . . 10 (πœ‘ β†’ 𝑉 ∈ π‘Š)
20 yoneda.v . . . . . . . . . . 11 (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
2120unssbd 4188 . . . . . . . . . 10 (πœ‘ β†’ π‘ˆ βŠ† 𝑉)
2219, 21ssexd 5324 . . . . . . . . 9 (πœ‘ β†’ π‘ˆ ∈ V)
23 yoneda.s . . . . . . . . . 10 𝑆 = (SetCatβ€˜π‘ˆ)
2423setccat 18032 . . . . . . . . 9 (π‘ˆ ∈ V β†’ 𝑆 ∈ Cat)
2522, 24syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑆 ∈ Cat)
268, 18, 25fuccat 17920 . . . . . . 7 (πœ‘ β†’ 𝑄 ∈ Cat)
27 eqid 2733 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
287, 26, 18, 272ndfcl 18147 . . . . . 6 (πœ‘ β†’ (𝑄 2ndF 𝑂) ∈ ((𝑄 Γ—c 𝑂) Func 𝑂))
29 eqid 2733 . . . . . . . 8 (oppCatβ€˜π‘„) = (oppCatβ€˜π‘„)
30 relfunc 17809 . . . . . . . . 9 Rel (𝐢 Func 𝑄)
31 yoneda.y . . . . . . . . . 10 π‘Œ = (Yonβ€˜πΆ)
32 yoneda.u . . . . . . . . . 10 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
3331, 16, 10, 23, 8, 22, 32yoncl 18212 . . . . . . . . 9 (πœ‘ β†’ π‘Œ ∈ (𝐢 Func 𝑄))
34 1st2ndbr 8025 . . . . . . . . 9 ((Rel (𝐢 Func 𝑄) ∧ π‘Œ ∈ (𝐢 Func 𝑄)) β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
3530, 33, 34sylancr 588 . . . . . . . 8 (πœ‘ β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
3610, 29, 35funcoppc 17822 . . . . . . 7 (πœ‘ β†’ (1st β€˜π‘Œ)(𝑂 Func (oppCatβ€˜π‘„))tpos (2nd β€˜π‘Œ))
37 df-br 5149 . . . . . . 7 ((1st β€˜π‘Œ)(𝑂 Func (oppCatβ€˜π‘„))tpos (2nd β€˜π‘Œ) ↔ ⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∈ (𝑂 Func (oppCatβ€˜π‘„)))
3836, 37sylib 217 . . . . . 6 (πœ‘ β†’ ⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∈ (𝑂 Func (oppCatβ€˜π‘„)))
3928, 38cofucl 17835 . . . . 5 (πœ‘ β†’ (⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 Γ—c 𝑂) Func (oppCatβ€˜π‘„)))
40 eqid 2733 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
417, 26, 18, 401stfcl 18146 . . . . 5 (πœ‘ β†’ (𝑄 1stF 𝑂) ∈ ((𝑄 Γ—c 𝑂) Func 𝑄))
4214, 15, 39, 41prfcl 18152 . . . 4 (πœ‘ β†’ ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) ∈ ((𝑄 Γ—c 𝑂) Func ((oppCatβ€˜π‘„) Γ—c 𝑄)))
43 yoneda.h . . . . 5 𝐻 = (HomFβ€˜π‘„)
44 yoneda.t . . . . 5 𝑇 = (SetCatβ€˜π‘‰)
4520unssad 4187 . . . . 5 (πœ‘ β†’ ran (Homf β€˜π‘„) βŠ† 𝑉)
4643, 29, 44, 26, 19, 45hofcl 18209 . . . 4 (πœ‘ β†’ 𝐻 ∈ (((oppCatβ€˜π‘„) Γ—c 𝑄) Func 𝑇))
47 yonedalem21.f . . . . 5 (πœ‘ β†’ 𝐹 ∈ (𝑂 Func 𝑆))
48 yonedalem21.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
4947, 48opelxpd 5714 . . . 4 (πœ‘ β†’ ⟨𝐹, π‘‹βŸ© ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
50 yonedalem22.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ (𝑂 Func 𝑆))
51 yonedalem22.p . . . . 5 (πœ‘ β†’ 𝑃 ∈ 𝐡)
5250, 51opelxpd 5714 . . . 4 (πœ‘ β†’ ⟨𝐺, π‘ƒβŸ© ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
53 eqid 2733 . . . 4 (Hom β€˜(𝑄 Γ—c 𝑂)) = (Hom β€˜(𝑄 Γ—c 𝑂))
54 yonedalem22.a . . . . . 6 (πœ‘ β†’ 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
55 yonedalem22.k . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋))
56 eqid 2733 . . . . . . . 8 (Hom β€˜πΆ) = (Hom β€˜πΆ)
5756, 10oppchom 17657 . . . . . . 7 (𝑋(Hom β€˜π‘‚)𝑃) = (𝑃(Hom β€˜πΆ)𝑋)
5855, 57eleqtrrdi 2845 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (𝑋(Hom β€˜π‘‚)𝑃))
5954, 58opelxpd 5714 . . . . 5 (πœ‘ β†’ ⟨𝐴, 𝐾⟩ ∈ ((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑋(Hom β€˜π‘‚)𝑃)))
60 eqid 2733 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
618, 60fuchom 17910 . . . . . 6 (𝑂 Nat 𝑆) = (Hom β€˜π‘„)
62 eqid 2733 . . . . . 6 (Hom β€˜π‘‚) = (Hom β€˜π‘‚)
637, 9, 12, 61, 62, 47, 48, 50, 51, 53xpchom2 18135 . . . . 5 (πœ‘ β†’ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©) = ((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑋(Hom β€˜π‘‚)𝑃)))
6459, 63eleqtrrd 2837 . . . 4 (πœ‘ β†’ ⟨𝐴, 𝐾⟩ ∈ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©))
6513, 42, 46, 49, 52, 53, 64cofu2 17833 . . 3 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = ((((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜π»)((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©))β€˜((⟨𝐹, π‘‹βŸ©(2nd β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)))
666, 65eqtrid 2785 . 2 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾) = ((((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜π»)((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©))β€˜((⟨𝐹, π‘‹βŸ©(2nd β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)))
6714, 13, 53, 39, 41, 49prf1 18149 . . . . . 6 (πœ‘ β†’ ((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©) = ⟨((1st β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©), ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)⟩)
6813, 28, 38, 49cofu1 17831 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©) = ((1st β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩)β€˜((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)))
69 fvex 6902 . . . . . . . . . . 11 (1st β€˜π‘Œ) ∈ V
70 fvex 6902 . . . . . . . . . . . 12 (2nd β€˜π‘Œ) ∈ V
7170tposex 8242 . . . . . . . . . . 11 tpos (2nd β€˜π‘Œ) ∈ V
7269, 71op1st 7980 . . . . . . . . . 10 (1st β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩) = (1st β€˜π‘Œ)
7372a1i 11 . . . . . . . . 9 (πœ‘ β†’ (1st β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩) = (1st β€˜π‘Œ))
747, 13, 53, 26, 18, 27, 492ndf1 18144 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = (2nd β€˜βŸ¨πΉ, π‘‹βŸ©))
75 op2ndg 7985 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐡) β†’ (2nd β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝑋)
7647, 48, 75syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ (2nd β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝑋)
7774, 76eqtrd 2773 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝑋)
7873, 77fveq12d 6896 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩)β€˜((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)) = ((1st β€˜π‘Œ)β€˜π‘‹))
7968, 78eqtrd 2773 . . . . . . 7 (πœ‘ β†’ ((1st β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©) = ((1st β€˜π‘Œ)β€˜π‘‹))
807, 13, 53, 26, 18, 40, 491stf1 18141 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = (1st β€˜βŸ¨πΉ, π‘‹βŸ©))
81 op1stg 7984 . . . . . . . . 9 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐡) β†’ (1st β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝐹)
8247, 48, 81syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (1st β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝐹)
8380, 82eqtrd 2773 . . . . . . 7 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝐹)
8479, 83opeq12d 4881 . . . . . 6 (πœ‘ β†’ ⟨((1st β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©), ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)⟩ = ⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩)
8567, 84eqtrd 2773 . . . . 5 (πœ‘ β†’ ((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©) = ⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩)
8614, 13, 53, 39, 41, 52prf1 18149 . . . . . 6 (πœ‘ β†’ ((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©) = ⟨((1st β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©), ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)⟩)
8713, 28, 38, 52cofu1 17831 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©) = ((1st β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩)β€˜((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)))
887, 13, 53, 26, 18, 27, 522ndf1 18144 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = (2nd β€˜βŸ¨πΊ, π‘ƒβŸ©))
89 op2ndg 7985 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃 ∈ 𝐡) β†’ (2nd β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝑃)
9050, 51, 89syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ (2nd β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝑃)
9188, 90eqtrd 2773 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝑃)
9273, 91fveq12d 6896 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩)β€˜((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)) = ((1st β€˜π‘Œ)β€˜π‘ƒ))
9387, 92eqtrd 2773 . . . . . . 7 (πœ‘ β†’ ((1st β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©) = ((1st β€˜π‘Œ)β€˜π‘ƒ))
947, 13, 53, 26, 18, 40, 521stf1 18141 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = (1st β€˜βŸ¨πΊ, π‘ƒβŸ©))
95 op1stg 7984 . . . . . . . . 9 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃 ∈ 𝐡) β†’ (1st β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝐺)
9650, 51, 95syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (1st β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝐺)
9794, 96eqtrd 2773 . . . . . . 7 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝐺)
9893, 97opeq12d 4881 . . . . . 6 (πœ‘ β†’ ⟨((1st β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©), ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)⟩ = ⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)
9986, 98eqtrd 2773 . . . . 5 (πœ‘ β†’ ((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©) = ⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)
10085, 99oveq12d 7424 . . . 4 (πœ‘ β†’ (((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜π»)((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©)) = (⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(2nd β€˜π»)⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩))
10114, 13, 53, 39, 41, 49, 52, 64prf2 18151 . . . . 5 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = ⟨((⟨𝐹, π‘‹βŸ©(2nd β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩), ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 1stF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)⟩)
10213, 28, 38, 49, 52, 53, 64cofu2 17833 . . . . . . 7 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = ((((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©))β€˜((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 2ndF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)))
10369, 71op2nd 7981 . . . . . . . . . . 11 (2nd β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩) = tpos (2nd β€˜π‘Œ)
104103oveqi 7419 . . . . . . . . . 10 (((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)) = (((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)tpos (2nd β€˜π‘Œ)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©))
105 ovtpos 8223 . . . . . . . . . 10 (((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)tpos (2nd β€˜π‘Œ)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)) = (((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)(2nd β€˜π‘Œ)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©))
106104, 105eqtri 2761 . . . . . . . . 9 (((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)) = (((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)(2nd β€˜π‘Œ)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©))
10791, 77oveq12d 7424 . . . . . . . . 9 (πœ‘ β†’ (((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)(2nd β€˜π‘Œ)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)) = (𝑃(2nd β€˜π‘Œ)𝑋))
108106, 107eqtrid 2785 . . . . . . . 8 (πœ‘ β†’ (((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©)) = (𝑃(2nd β€˜π‘Œ)𝑋))
1097, 13, 53, 26, 18, 27, 49, 522ndf2 18145 . . . . . . . . . 10 (πœ‘ β†’ (⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 2ndF 𝑂))⟨𝐺, π‘ƒβŸ©) = (2nd β†Ύ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©)))
110109fveq1d 6891 . . . . . . . . 9 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 2ndF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = ((2nd β†Ύ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©))β€˜βŸ¨π΄, 𝐾⟩))
11164fvresd 6909 . . . . . . . . 9 (πœ‘ β†’ ((2nd β†Ύ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©))β€˜βŸ¨π΄, 𝐾⟩) = (2nd β€˜βŸ¨π΄, 𝐾⟩))
112 op2ndg 7985 . . . . . . . . . 10 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋)) β†’ (2nd β€˜βŸ¨π΄, 𝐾⟩) = 𝐾)
11354, 55, 112syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜βŸ¨π΄, 𝐾⟩) = 𝐾)
114110, 111, 1133eqtrd 2777 . . . . . . . 8 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 2ndF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = 𝐾)
115108, 114fveq12d 6896 . . . . . . 7 (πœ‘ β†’ ((((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩)((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©))β€˜((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 2ndF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)) = ((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))
116102, 115eqtrd 2773 . . . . . 6 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = ((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))
1177, 13, 53, 26, 18, 40, 49, 521stf2 18142 . . . . . . . 8 (πœ‘ β†’ (⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 1stF 𝑂))⟨𝐺, π‘ƒβŸ©) = (1st β†Ύ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©)))
118117fveq1d 6891 . . . . . . 7 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 1stF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = ((1st β†Ύ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©))β€˜βŸ¨π΄, 𝐾⟩))
11964fvresd 6909 . . . . . . 7 (πœ‘ β†’ ((1st β†Ύ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©))β€˜βŸ¨π΄, 𝐾⟩) = (1st β€˜βŸ¨π΄, 𝐾⟩))
120 op1stg 7984 . . . . . . . 8 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋)) β†’ (1st β€˜βŸ¨π΄, 𝐾⟩) = 𝐴)
12154, 55, 120syl2anc 585 . . . . . . 7 (πœ‘ β†’ (1st β€˜βŸ¨π΄, 𝐾⟩) = 𝐴)
122118, 119, 1213eqtrd 2777 . . . . . 6 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 1stF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = 𝐴)
123116, 122opeq12d 4881 . . . . 5 (πœ‘ β†’ ⟨((⟨𝐹, π‘‹βŸ©(2nd β€˜(⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩), ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 1stF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)⟩ = ⟨((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ), 𝐴⟩)
124101, 123eqtrd 2773 . . . 4 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = ⟨((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ), 𝐴⟩)
125100, 124fveq12d 6896 . . 3 (πœ‘ β†’ ((((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜π»)((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©))β€˜((⟨𝐹, π‘‹βŸ©(2nd β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)) = ((⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(2nd β€˜π»)⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)β€˜βŸ¨((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ), 𝐴⟩))
126 df-ov 7409 . . 3 (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(2nd β€˜π»)⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)𝐴) = ((⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(2nd β€˜π»)⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)β€˜βŸ¨((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ), 𝐴⟩)
127125, 126eqtr4di 2791 . 2 (πœ‘ β†’ ((((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΉ, π‘‹βŸ©)(2nd β€˜π»)((1st β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))β€˜βŸ¨πΊ, π‘ƒβŸ©))β€˜((⟨𝐹, π‘‹βŸ©(2nd β€˜((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩)) = (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(2nd β€˜π»)⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)𝐴))
12866, 127eqtrd 2773 1 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾) = (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(2nd β€˜π»)⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475   βˆͺ cun 3946   βŠ† wss 3948  βŸ¨cop 4634   class class class wbr 5148   Γ— cxp 5674  ran crn 5677   β†Ύ cres 5678  Rel wrel 5681  β€˜cfv 6541  (class class class)co 7406  1st c1st 7970  2nd c2nd 7971  tpos ctpos 8207  Basecbs 17141  Hom chom 17205  Catccat 17605  Idccid 17606  Homf chomf 17607  oppCatcoppc 17652   Func cfunc 17801   ∘func ccofu 17803   Nat cnat 17889   FuncCat cfuc 17890  SetCatcsetc 18022   Γ—c cxpc 18117   1stF c1stf 18118   2ndF c2ndf 18119   ⟨,⟩F cprf 18120   evalF cevlf 18159  HomFchof 18198  Yoncyon 18199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-hom 17218  df-cco 17219  df-cat 17609  df-cid 17610  df-homf 17611  df-comf 17612  df-oppc 17653  df-func 17805  df-cofu 17807  df-nat 17891  df-fuc 17892  df-setc 18023  df-xpc 18121  df-1stf 18122  df-2ndf 18123  df-prf 18124  df-curf 18164  df-hof 18200  df-yon 18201
This theorem is referenced by:  yonedalem3b  18229
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