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Theorem yonedalem3a 18251
Description: Lemma for yoneda 18260. (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)
yonedalem3a.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))))
Assertion
Ref Expression
yonedalem3a (πœ‘ β†’ ((𝐹𝑀𝑋) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))) ∧ (𝐹𝑀𝑋):(𝐹(1st β€˜π‘)𝑋)⟢(𝐹(1st β€˜πΈ)𝑋)))
Distinct variable groups:   𝑓,π‘Ž,π‘₯, 1   𝐢,π‘Ž,𝑓,π‘₯   𝐸,π‘Ž,𝑓   𝐹,π‘Ž,𝑓,π‘₯   𝐡,π‘Ž,𝑓,π‘₯   𝑂,π‘Ž,𝑓,π‘₯   𝑆,π‘Ž,𝑓,π‘₯   𝑄,π‘Ž,𝑓,π‘₯   𝑇,𝑓   πœ‘,π‘Ž,𝑓,π‘₯   π‘Œ,π‘Ž,𝑓,π‘₯   𝑍,π‘Ž,𝑓,π‘₯   𝑋,π‘Ž,𝑓,π‘₯
Allowed substitution hints:   𝑅(π‘₯,𝑓,π‘Ž)   𝑇(π‘₯,π‘Ž)   π‘ˆ(π‘₯,𝑓,π‘Ž)   𝐸(π‘₯)   𝐻(π‘₯,𝑓,π‘Ž)   𝑀(π‘₯,𝑓,π‘Ž)   𝑉(π‘₯,𝑓,π‘Ž)   π‘Š(π‘₯,𝑓,π‘Ž)

Proof of Theorem yonedalem3a
StepHypRef Expression
1 yonedalem21.f . . 3 (πœ‘ β†’ 𝐹 ∈ (𝑂 Func 𝑆))
2 yonedalem21.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
3 simpr 484 . . . . . . 7 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ π‘₯ = 𝑋)
43fveq2d 6895 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ ((1st β€˜π‘Œ)β€˜π‘₯) = ((1st β€˜π‘Œ)β€˜π‘‹))
5 simpl 482 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ 𝑓 = 𝐹)
64, 5oveq12d 7432 . . . . 5 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) = (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹))
73fveq2d 6895 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ (π‘Žβ€˜π‘₯) = (π‘Žβ€˜π‘‹))
83fveq2d 6895 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ ( 1 β€˜π‘₯) = ( 1 β€˜π‘‹))
97, 8fveq12d 6898 . . . . 5 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯)) = ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)))
106, 9mpteq12dv 5233 . . . 4 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))
11 yonedalem3a.m . . . 4 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))))
12 ovex 7447 . . . . 5 (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ∈ V
1312mptex 7229 . . . 4 (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))) ∈ V
1410, 11, 13ovmpoa 7568 . . 3 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐡) β†’ (𝐹𝑀𝑋) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))
151, 2, 14syl2anc 583 . 2 (πœ‘ β†’ (𝐹𝑀𝑋) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))
16 eqid 2727 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
17 simpr 484 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹))
1816, 17nat1st2nd 17926 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ π‘Ž ∈ (⟨(1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)), (2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))⟩(𝑂 Nat 𝑆)⟨(1st β€˜πΉ), (2nd β€˜πΉ)⟩))
19 yoneda.o . . . . . . . 8 𝑂 = (oppCatβ€˜πΆ)
20 yoneda.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
2119, 20oppcbas 17684 . . . . . . 7 𝐡 = (Baseβ€˜π‘‚)
22 eqid 2727 . . . . . . 7 (Hom β€˜π‘†) = (Hom β€˜π‘†)
232adantr 480 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝑋 ∈ 𝐡)
2416, 18, 21, 22, 23natcl 17928 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (π‘Žβ€˜π‘‹) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜πΉ)β€˜π‘‹)))
25 yoneda.s . . . . . . 7 𝑆 = (SetCatβ€˜π‘ˆ)
26 yoneda.w . . . . . . . . 9 (πœ‘ β†’ 𝑉 ∈ π‘Š)
27 yoneda.v . . . . . . . . . 10 (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
2827unssbd 4184 . . . . . . . . 9 (πœ‘ β†’ π‘ˆ βŠ† 𝑉)
2926, 28ssexd 5318 . . . . . . . 8 (πœ‘ β†’ π‘ˆ ∈ V)
3029adantr 480 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ π‘ˆ ∈ V)
31 eqid 2727 . . . . . . . . . . 11 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
32 relfunc 17833 . . . . . . . . . . . 12 Rel (𝑂 Func 𝑆)
33 yoneda.y . . . . . . . . . . . . 13 π‘Œ = (Yonβ€˜πΆ)
34 yoneda.c . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐢 ∈ Cat)
35 yoneda.u . . . . . . . . . . . . 13 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
3633, 20, 34, 2, 19, 25, 29, 35yon1cl 18240 . . . . . . . . . . . 12 (πœ‘ β†’ ((1st β€˜π‘Œ)β€˜π‘‹) ∈ (𝑂 Func 𝑆))
37 1st2ndbr 8038 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ ((1st β€˜π‘Œ)β€˜π‘‹) ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))(𝑂 Func 𝑆)(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹)))
3832, 36, 37sylancr 586 . . . . . . . . . . 11 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))(𝑂 Func 𝑆)(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹)))
3921, 31, 38funcf1 17837 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)):𝐡⟢(Baseβ€˜π‘†))
4039, 2ffvelcdmd 7089 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) ∈ (Baseβ€˜π‘†))
4125, 29setcbas 18052 . . . . . . . . 9 (πœ‘ β†’ π‘ˆ = (Baseβ€˜π‘†))
4240, 41eleqtrrd 2831 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) ∈ π‘ˆ)
4342adantr 480 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) ∈ π‘ˆ)
44 1st2ndbr 8038 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜πΉ)(𝑂 Func 𝑆)(2nd β€˜πΉ))
4532, 1, 44sylancr 586 . . . . . . . . . . 11 (πœ‘ β†’ (1st β€˜πΉ)(𝑂 Func 𝑆)(2nd β€˜πΉ))
4621, 31, 45funcf1 17837 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π‘†))
4746, 2ffvelcdmd 7089 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜πΉ)β€˜π‘‹) ∈ (Baseβ€˜π‘†))
4847, 41eleqtrrd 2831 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜πΉ)β€˜π‘‹) ∈ π‘ˆ)
4948adantr 480 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((1st β€˜πΉ)β€˜π‘‹) ∈ π‘ˆ)
5025, 30, 22, 43, 49elsetchom 18055 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘‹) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜πΉ)β€˜π‘‹)) ↔ (π‘Žβ€˜π‘‹):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)⟢((1st β€˜πΉ)β€˜π‘‹)))
5124, 50mpbid 231 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (π‘Žβ€˜π‘‹):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)⟢((1st β€˜πΉ)β€˜π‘‹))
52 eqid 2727 . . . . . . . 8 (Hom β€˜πΆ) = (Hom β€˜πΆ)
53 yoneda.1 . . . . . . . 8 1 = (Idβ€˜πΆ)
5420, 52, 53, 34, 2catidcl 17647 . . . . . . 7 (πœ‘ β†’ ( 1 β€˜π‘‹) ∈ (𝑋(Hom β€˜πΆ)𝑋))
5533, 20, 34, 2, 52, 2yon11 18241 . . . . . . 7 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) = (𝑋(Hom β€˜πΆ)𝑋))
5654, 55eleqtrrd 2831 . . . . . 6 (πœ‘ β†’ ( 1 β€˜π‘‹) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹))
5756adantr 480 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ( 1 β€˜π‘‹) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹))
5851, 57ffvelcdmd 7089 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)) ∈ ((1st β€˜πΉ)β€˜π‘‹))
5958fmpttd 7119 . . 3 (πœ‘ β†’ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))):(((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)⟢((1st β€˜πΉ)β€˜π‘‹))
60 yoneda.t . . . . 5 𝑇 = (SetCatβ€˜π‘‰)
61 yoneda.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
62 yoneda.h . . . . 5 𝐻 = (HomFβ€˜π‘„)
63 yoneda.r . . . . 5 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
64 yoneda.e . . . . 5 𝐸 = (𝑂 evalF 𝑆)
65 yoneda.z . . . . 5 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
6633, 20, 53, 19, 25, 60, 61, 62, 63, 64, 65, 34, 26, 35, 27, 1, 2yonedalem21 18250 . . . 4 (πœ‘ β†’ (𝐹(1st β€˜π‘)𝑋) = (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹))
6719oppccat 17689 . . . . . 6 (𝐢 ∈ Cat β†’ 𝑂 ∈ Cat)
6834, 67syl 17 . . . . 5 (πœ‘ β†’ 𝑂 ∈ Cat)
6925setccat 18059 . . . . . 6 (π‘ˆ ∈ V β†’ 𝑆 ∈ Cat)
7029, 69syl 17 . . . . 5 (πœ‘ β†’ 𝑆 ∈ Cat)
7164, 68, 70, 21, 1, 2evlf1 18197 . . . 4 (πœ‘ β†’ (𝐹(1st β€˜πΈ)𝑋) = ((1st β€˜πΉ)β€˜π‘‹))
7215, 66, 71feq123d 6705 . . 3 (πœ‘ β†’ ((𝐹𝑀𝑋):(𝐹(1st β€˜π‘)𝑋)⟢(𝐹(1st β€˜πΈ)𝑋) ↔ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))):(((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)⟢((1st β€˜πΉ)β€˜π‘‹)))
7359, 72mpbird 257 . 2 (πœ‘ β†’ (𝐹𝑀𝑋):(𝐹(1st β€˜π‘)𝑋)⟢(𝐹(1st β€˜πΈ)𝑋))
7415, 73jca 511 1 (πœ‘ β†’ ((𝐹𝑀𝑋) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))) ∧ (𝐹𝑀𝑋):(𝐹(1st β€˜π‘)𝑋)⟢(𝐹(1st β€˜πΈ)𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3469   βˆͺ cun 3942   βŠ† wss 3944  βŸ¨cop 4630   class class class wbr 5142   ↦ cmpt 5225  ran crn 5673  Rel wrel 5677  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  1st c1st 7983  2nd c2nd 7984  tpos ctpos 8222  Basecbs 17165  Hom chom 17229  Catccat 17629  Idccid 17630  Homf chomf 17631  oppCatcoppc 17676   Func cfunc 17825   ∘func ccofu 17827   Nat cnat 17916   FuncCat cfuc 17917  SetCatcsetc 18049   Γ—c cxpc 18144   1stF c1stf 18145   2ndF c2ndf 18146   ⟨,⟩F cprf 18147   evalF cevlf 18186  HomFchof 18225  Yoncyon 18226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-tpos 8223  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-ixp 8906  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12489  df-z 12575  df-dec 12694  df-uz 12839  df-fz 13503  df-struct 17101  df-sets 17118  df-slot 17136  df-ndx 17148  df-base 17166  df-hom 17242  df-cco 17243  df-cat 17633  df-cid 17634  df-homf 17635  df-comf 17636  df-oppc 17677  df-func 17829  df-cofu 17831  df-nat 17918  df-fuc 17919  df-setc 18050  df-xpc 18148  df-1stf 18149  df-2ndf 18150  df-prf 18151  df-evlf 18190  df-curf 18191  df-hof 18227  df-yon 18228
This theorem is referenced by:  yonedalem3b  18256  yonedalem3  18257  yonedainv  18258
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