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Theorem yonedalem22 18233
Description: Lemma for yoneda 18238. (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 6884 . . . . . 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 7414 . . . . 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 7414 . . . 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 7404 . . . 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 2752 . . 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 2724 . . . . 5 (𝑄 Γ—c 𝑂) = (𝑄 Γ—c 𝑂)
8 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
98fucbas 17914 . . . . 5 (𝑂 Func 𝑆) = (Baseβ€˜π‘„)
10 yoneda.o . . . . . 6 𝑂 = (oppCatβ€˜πΆ)
11 yoneda.b . . . . . 6 𝐡 = (Baseβ€˜πΆ)
1210, 11oppcbas 17662 . . . . 5 𝐡 = (Baseβ€˜π‘‚)
137, 9, 12xpcbas 18132 . . . 4 ((𝑂 Func 𝑆) Γ— 𝐡) = (Baseβ€˜(𝑄 Γ—c 𝑂))
14 eqid 2724 . . . . 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 2724 . . . . 5 ((oppCatβ€˜π‘„) Γ—c 𝑄) = ((oppCatβ€˜π‘„) Γ—c 𝑄)
16 yoneda.c . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ Cat)
1710oppccat 17667 . . . . . . . . 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 4180 . . . . . . . . . 10 (πœ‘ β†’ π‘ˆ βŠ† 𝑉)
2219, 21ssexd 5314 . . . . . . . . 9 (πœ‘ β†’ π‘ˆ ∈ V)
23 yoneda.s . . . . . . . . . 10 𝑆 = (SetCatβ€˜π‘ˆ)
2423setccat 18037 . . . . . . . . 9 (π‘ˆ ∈ V β†’ 𝑆 ∈ Cat)
2522, 24syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑆 ∈ Cat)
268, 18, 25fuccat 17925 . . . . . . 7 (πœ‘ β†’ 𝑄 ∈ Cat)
27 eqid 2724 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
287, 26, 18, 272ndfcl 18152 . . . . . 6 (πœ‘ β†’ (𝑄 2ndF 𝑂) ∈ ((𝑄 Γ—c 𝑂) Func 𝑂))
29 eqid 2724 . . . . . . . 8 (oppCatβ€˜π‘„) = (oppCatβ€˜π‘„)
30 relfunc 17811 . . . . . . . . 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 18217 . . . . . . . . 9 (πœ‘ β†’ π‘Œ ∈ (𝐢 Func 𝑄))
34 1st2ndbr 8021 . . . . . . . . 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 586 . . . . . . . 8 (πœ‘ β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
3610, 29, 35funcoppc 17824 . . . . . . 7 (πœ‘ β†’ (1st β€˜π‘Œ)(𝑂 Func (oppCatβ€˜π‘„))tpos (2nd β€˜π‘Œ))
37 df-br 5139 . . . . . . 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 17837 . . . . 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 2724 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
417, 26, 18, 401stfcl 18151 . . . . 5 (πœ‘ β†’ (𝑄 1stF 𝑂) ∈ ((𝑄 Γ—c 𝑂) Func 𝑄))
4214, 15, 39, 41prfcl 18157 . . . 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 4179 . . . . 5 (πœ‘ β†’ ran (Homf β€˜π‘„) βŠ† 𝑉)
4643, 29, 44, 26, 19, 45hofcl 18214 . . . 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 5705 . . . 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 5705 . . . 4 (πœ‘ β†’ ⟨𝐺, π‘ƒβŸ© ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
53 eqid 2724 . . . 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 2724 . . . . . . . 8 (Hom β€˜πΆ) = (Hom β€˜πΆ)
5756, 10oppchom 17659 . . . . . . 7 (𝑋(Hom β€˜π‘‚)𝑃) = (𝑃(Hom β€˜πΆ)𝑋)
5855, 57eleqtrrdi 2836 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (𝑋(Hom β€˜π‘‚)𝑃))
5954, 58opelxpd 5705 . . . . 5 (πœ‘ β†’ ⟨𝐴, 𝐾⟩ ∈ ((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑋(Hom β€˜π‘‚)𝑃)))
60 eqid 2724 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
618, 60fuchom 17915 . . . . . 6 (𝑂 Nat 𝑆) = (Hom β€˜π‘„)
62 eqid 2724 . . . . . 6 (Hom β€˜π‘‚) = (Hom β€˜π‘‚)
637, 9, 12, 61, 62, 47, 48, 50, 51, 53xpchom2 18140 . . . . 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 2828 . . . 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 17835 . . 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 2776 . 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 18154 . . . . . 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 17833 . . . . . . . 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 6894 . . . . . . . . . . 11 (1st β€˜π‘Œ) ∈ V
70 fvex 6894 . . . . . . . . . . . 12 (2nd β€˜π‘Œ) ∈ V
7170tposex 8240 . . . . . . . . . . 11 tpos (2nd β€˜π‘Œ) ∈ V
7269, 71op1st 7976 . . . . . . . . . 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 18149 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = (2nd β€˜βŸ¨πΉ, π‘‹βŸ©))
75 op2ndg 7981 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐡) β†’ (2nd β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝑋)
7647, 48, 75syl2anc 583 . . . . . . . . . 10 (πœ‘ β†’ (2nd β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝑋)
7774, 76eqtrd 2764 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝑋)
7873, 77fveq12d 6888 . . . . . . . 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 2764 . . . . . . 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 18146 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = (1st β€˜βŸ¨πΉ, π‘‹βŸ©))
81 op1stg 7980 . . . . . . . . 9 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐡) β†’ (1st β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝐹)
8247, 48, 81syl2anc 583 . . . . . . . 8 (πœ‘ β†’ (1st β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝐹)
8380, 82eqtrd 2764 . . . . . . 7 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝐹)
8479, 83opeq12d 4873 . . . . . 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 2764 . . . . 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 18154 . . . . . 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 17833 . . . . . . . 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 18149 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = (2nd β€˜βŸ¨πΊ, π‘ƒβŸ©))
89 op2ndg 7981 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃 ∈ 𝐡) β†’ (2nd β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝑃)
9050, 51, 89syl2anc 583 . . . . . . . . . 10 (πœ‘ β†’ (2nd β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝑃)
9188, 90eqtrd 2764 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝑃)
9273, 91fveq12d 6888 . . . . . . . 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 2764 . . . . . . 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 18146 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = (1st β€˜βŸ¨πΊ, π‘ƒβŸ©))
95 op1stg 7980 . . . . . . . . 9 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃 ∈ 𝐡) β†’ (1st β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝐺)
9650, 51, 95syl2anc 583 . . . . . . . 8 (πœ‘ β†’ (1st β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝐺)
9794, 96eqtrd 2764 . . . . . . 7 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝐺)
9893, 97opeq12d 4873 . . . . . 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 2764 . . . . 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 7419 . . . 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 18156 . . . . 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 17835 . . . . . . 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 7977 . . . . . . . . . . 11 (2nd β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩) = tpos (2nd β€˜π‘Œ)
104103oveqi 7414 . . . . . . . . . 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 8221 . . . . . . . . . 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 2752 . . . . . . . . 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 7419 . . . . . . . . 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 2776 . . . . . . . 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 18150 . . . . . . . . . 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 6883 . . . . . . . . 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 6901 . . . . . . . . 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 7981 . . . . . . . . . 10 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋)) β†’ (2nd β€˜βŸ¨π΄, 𝐾⟩) = 𝐾)
11354, 55, 112syl2anc 583 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜βŸ¨π΄, 𝐾⟩) = 𝐾)
114110, 111, 1133eqtrd 2768 . . . . . . . 8 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 2ndF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = 𝐾)
115108, 114fveq12d 6888 . . . . . . 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 2764 . . . . . 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 18147 . . . . . . . 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 6883 . . . . . . 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 6901 . . . . . . 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 7980 . . . . . . . 8 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋)) β†’ (1st β€˜βŸ¨π΄, 𝐾⟩) = 𝐴)
12154, 55, 120syl2anc 583 . . . . . . 7 (πœ‘ β†’ (1st β€˜βŸ¨π΄, 𝐾⟩) = 𝐴)
122118, 119, 1213eqtrd 2768 . . . . . 6 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 1stF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = 𝐴)
123116, 122opeq12d 4873 . . . . 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 2764 . . . 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 6888 . . 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 7404 . . 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 2782 . 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 2764 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 1533   ∈ wcel 2098  Vcvv 3466   βˆͺ cun 3938   βŠ† wss 3940  βŸ¨cop 4626   class class class wbr 5138   Γ— cxp 5664  ran crn 5667   β†Ύ cres 5668  Rel wrel 5671  β€˜cfv 6533  (class class class)co 7401  1st c1st 7966  2nd c2nd 7967  tpos ctpos 8205  Basecbs 17143  Hom chom 17207  Catccat 17607  Idccid 17608  Homf chomf 17609  oppCatcoppc 17654   Func cfunc 17803   ∘func ccofu 17805   Nat cnat 17894   FuncCat cfuc 17895  SetCatcsetc 18027   Γ—c cxpc 18122   1stF c1stf 18123   2ndF c2ndf 18124   ⟨,⟩F cprf 18125   evalF cevlf 18164  HomFchof 18203  Yoncyon 18204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  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 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-hom 17220  df-cco 17221  df-cat 17611  df-cid 17612  df-homf 17613  df-comf 17614  df-oppc 17655  df-func 17807  df-cofu 17809  df-nat 17896  df-fuc 17897  df-setc 18028  df-xpc 18126  df-1stf 18127  df-2ndf 18128  df-prf 18129  df-curf 18169  df-hof 18205  df-yon 18206
This theorem is referenced by:  yonedalem3b  18234
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