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Theorem yonedalem22 18235
Description: Lemma for yoneda 18240. (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 6893 . . . . . 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 7424 . . . . 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 7424 . . . 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 7414 . . . 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 2758 . . 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 2730 . . . . 5 (𝑄 Γ—c 𝑂) = (𝑄 Γ—c 𝑂)
8 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
98fucbas 17916 . . . . 5 (𝑂 Func 𝑆) = (Baseβ€˜π‘„)
10 yoneda.o . . . . . 6 𝑂 = (oppCatβ€˜πΆ)
11 yoneda.b . . . . . 6 𝐡 = (Baseβ€˜πΆ)
1210, 11oppcbas 17667 . . . . 5 𝐡 = (Baseβ€˜π‘‚)
137, 9, 12xpcbas 18134 . . . 4 ((𝑂 Func 𝑆) Γ— 𝐡) = (Baseβ€˜(𝑄 Γ—c 𝑂))
14 eqid 2730 . . . . 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 2730 . . . . 5 ((oppCatβ€˜π‘„) Γ—c 𝑄) = ((oppCatβ€˜π‘„) Γ—c 𝑄)
16 yoneda.c . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ Cat)
1710oppccat 17672 . . . . . . . . 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 4187 . . . . . . . . . 10 (πœ‘ β†’ π‘ˆ βŠ† 𝑉)
2219, 21ssexd 5323 . . . . . . . . 9 (πœ‘ β†’ π‘ˆ ∈ V)
23 yoneda.s . . . . . . . . . 10 𝑆 = (SetCatβ€˜π‘ˆ)
2423setccat 18039 . . . . . . . . 9 (π‘ˆ ∈ V β†’ 𝑆 ∈ Cat)
2522, 24syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑆 ∈ Cat)
268, 18, 25fuccat 17927 . . . . . . 7 (πœ‘ β†’ 𝑄 ∈ Cat)
27 eqid 2730 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
287, 26, 18, 272ndfcl 18154 . . . . . 6 (πœ‘ β†’ (𝑄 2ndF 𝑂) ∈ ((𝑄 Γ—c 𝑂) Func 𝑂))
29 eqid 2730 . . . . . . . 8 (oppCatβ€˜π‘„) = (oppCatβ€˜π‘„)
30 relfunc 17816 . . . . . . . . 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 18219 . . . . . . . . 9 (πœ‘ β†’ π‘Œ ∈ (𝐢 Func 𝑄))
34 1st2ndbr 8030 . . . . . . . . 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 585 . . . . . . . 8 (πœ‘ β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
3610, 29, 35funcoppc 17829 . . . . . . 7 (πœ‘ β†’ (1st β€˜π‘Œ)(𝑂 Func (oppCatβ€˜π‘„))tpos (2nd β€˜π‘Œ))
37 df-br 5148 . . . . . . 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 17842 . . . . 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 2730 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
417, 26, 18, 401stfcl 18153 . . . . 5 (πœ‘ β†’ (𝑄 1stF 𝑂) ∈ ((𝑄 Γ—c 𝑂) Func 𝑄))
4214, 15, 39, 41prfcl 18159 . . . 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 4186 . . . . 5 (πœ‘ β†’ ran (Homf β€˜π‘„) βŠ† 𝑉)
4643, 29, 44, 26, 19, 45hofcl 18216 . . . 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 2730 . . . 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 2730 . . . . . . . 8 (Hom β€˜πΆ) = (Hom β€˜πΆ)
5756, 10oppchom 17664 . . . . . . 7 (𝑋(Hom β€˜π‘‚)𝑃) = (𝑃(Hom β€˜πΆ)𝑋)
5855, 57eleqtrrdi 2842 . . . . . 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 2730 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
618, 60fuchom 17917 . . . . . 6 (𝑂 Nat 𝑆) = (Hom β€˜π‘„)
62 eqid 2730 . . . . . 6 (Hom β€˜π‘‚) = (Hom β€˜π‘‚)
637, 9, 12, 61, 62, 47, 48, 50, 51, 53xpchom2 18142 . . . . 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 2834 . . . 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 17840 . . 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 2782 . 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 18156 . . . . . 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 17838 . . . . . . . 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 6903 . . . . . . . . . . 11 (1st β€˜π‘Œ) ∈ V
70 fvex 6903 . . . . . . . . . . . 12 (2nd β€˜π‘Œ) ∈ V
7170tposex 8247 . . . . . . . . . . 11 tpos (2nd β€˜π‘Œ) ∈ V
7269, 71op1st 7985 . . . . . . . . . 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 18151 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = (2nd β€˜βŸ¨πΉ, π‘‹βŸ©))
75 op2ndg 7990 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐡) β†’ (2nd β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝑋)
7647, 48, 75syl2anc 582 . . . . . . . . . 10 (πœ‘ β†’ (2nd β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝑋)
7774, 76eqtrd 2770 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝑋)
7873, 77fveq12d 6897 . . . . . . . 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 2770 . . . . . . 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 18148 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = (1st β€˜βŸ¨πΉ, π‘‹βŸ©))
81 op1stg 7989 . . . . . . . . 9 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐡) β†’ (1st β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝐹)
8247, 48, 81syl2anc 582 . . . . . . . 8 (πœ‘ β†’ (1st β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝐹)
8380, 82eqtrd 2770 . . . . . . 7 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΉ, π‘‹βŸ©) = 𝐹)
8479, 83opeq12d 4880 . . . . . 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 2770 . . . . 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 18156 . . . . . 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 17838 . . . . . . . 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 18151 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = (2nd β€˜βŸ¨πΊ, π‘ƒβŸ©))
89 op2ndg 7990 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃 ∈ 𝐡) β†’ (2nd β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝑃)
9050, 51, 89syl2anc 582 . . . . . . . . . 10 (πœ‘ β†’ (2nd β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝑃)
9188, 90eqtrd 2770 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜(𝑄 2ndF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝑃)
9273, 91fveq12d 6897 . . . . . . . 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 2770 . . . . . . 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 18148 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = (1st β€˜βŸ¨πΊ, π‘ƒβŸ©))
95 op1stg 7989 . . . . . . . . 9 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃 ∈ 𝐡) β†’ (1st β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝐺)
9650, 51, 95syl2anc 582 . . . . . . . 8 (πœ‘ β†’ (1st β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝐺)
9794, 96eqtrd 2770 . . . . . . 7 (πœ‘ β†’ ((1st β€˜(𝑄 1stF 𝑂))β€˜βŸ¨πΊ, π‘ƒβŸ©) = 𝐺)
9893, 97opeq12d 4880 . . . . . 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 2770 . . . . 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 7429 . . . 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 18158 . . . . 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 17840 . . . . . . 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 7986 . . . . . . . . . . 11 (2nd β€˜βŸ¨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩) = tpos (2nd β€˜π‘Œ)
104103oveqi 7424 . . . . . . . . . 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 8228 . . . . . . . . . 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 2758 . . . . . . . . 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 7429 . . . . . . . . 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 2782 . . . . . . . 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 18152 . . . . . . . . . 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 6892 . . . . . . . . 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 6910 . . . . . . . . 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 7990 . . . . . . . . . 10 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋)) β†’ (2nd β€˜βŸ¨π΄, 𝐾⟩) = 𝐾)
11354, 55, 112syl2anc 582 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜βŸ¨π΄, 𝐾⟩) = 𝐾)
114110, 111, 1133eqtrd 2774 . . . . . . . 8 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 2ndF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = 𝐾)
115108, 114fveq12d 6897 . . . . . . 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 2770 . . . . . 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 18149 . . . . . . . 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 6892 . . . . . . 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 6910 . . . . . . 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 7989 . . . . . . . 8 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋)) β†’ (1st β€˜βŸ¨π΄, 𝐾⟩) = 𝐴)
12154, 55, 120syl2anc 582 . . . . . . 7 (πœ‘ β†’ (1st β€˜βŸ¨π΄, 𝐾⟩) = 𝐴)
122118, 119, 1213eqtrd 2774 . . . . . 6 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜(𝑄 1stF 𝑂))⟨𝐺, π‘ƒβŸ©)β€˜βŸ¨π΄, 𝐾⟩) = 𝐴)
123116, 122opeq12d 4880 . . . . 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 2770 . . . 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 6897 . . 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 7414 . . 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 2788 . 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 2770 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 1539   ∈ wcel 2104  Vcvv 3472   βˆͺ cun 3945   βŠ† wss 3947  βŸ¨cop 4633   class class class wbr 5147   Γ— cxp 5673  ran crn 5676   β†Ύ cres 5677  Rel wrel 5680  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  tpos ctpos 8212  Basecbs 17148  Hom chom 17212  Catccat 17612  Idccid 17613  Homf chomf 17614  oppCatcoppc 17659   Func cfunc 17808   ∘func ccofu 17810   Nat cnat 17896   FuncCat cfuc 17897  SetCatcsetc 18029   Γ—c cxpc 18124   1stF c1stf 18125   2ndF c2ndf 18126   ⟨,⟩F cprf 18127   evalF cevlf 18166  HomFchof 18205  Yoncyon 18206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-hom 17225  df-cco 17226  df-cat 17616  df-cid 17617  df-homf 17618  df-comf 17619  df-oppc 17660  df-func 17812  df-cofu 17814  df-nat 17898  df-fuc 17899  df-setc 18030  df-xpc 18128  df-1stf 18129  df-2ndf 18130  df-prf 18131  df-curf 18171  df-hof 18207  df-yon 18208
This theorem is referenced by:  yonedalem3b  18236
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