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Theorem yonedalem3a 18260
Description: Lemma for yoneda 18269. (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 483 . . . . . . 7 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ π‘₯ = 𝑋)
43fveq2d 6894 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ ((1st β€˜π‘Œ)β€˜π‘₯) = ((1st β€˜π‘Œ)β€˜π‘‹))
5 simpl 481 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ 𝑓 = 𝐹)
64, 5oveq12d 7431 . . . . 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 6894 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ (π‘Žβ€˜π‘₯) = (π‘Žβ€˜π‘‹))
83fveq2d 6894 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ ( 1 β€˜π‘₯) = ( 1 β€˜π‘‹))
97, 8fveq12d 6897 . . . . 5 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑋) β†’ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯)) = ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)))
106, 9mpteq12dv 5235 . . . 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 7446 . . . . 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 7570 . . 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 582 . 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 2725 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
17 simpr 483 . . . . . . . 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 17935 . . . . . . 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 17693 . . . . . . 7 𝐡 = (Baseβ€˜π‘‚)
22 eqid 2725 . . . . . . 7 (Hom β€˜π‘†) = (Hom β€˜π‘†)
232adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝑋 ∈ 𝐡)
2416, 18, 21, 22, 23natcl 17937 . . . . . 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 4183 . . . . . . . . 9 (πœ‘ β†’ π‘ˆ βŠ† 𝑉)
2926, 28ssexd 5320 . . . . . . . 8 (πœ‘ β†’ π‘ˆ ∈ V)
3029adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ π‘ˆ ∈ V)
31 eqid 2725 . . . . . . . . . . 11 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
32 relfunc 17842 . . . . . . . . . . . 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 18249 . . . . . . . . . . . 12 (πœ‘ β†’ ((1st β€˜π‘Œ)β€˜π‘‹) ∈ (𝑂 Func 𝑆))
37 1st2ndbr 8040 . . . . . . . . . . . 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 585 . . . . . . . . . . 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 17846 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)):𝐡⟢(Baseβ€˜π‘†))
4039, 2ffvelcdmd 7088 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) ∈ (Baseβ€˜π‘†))
4125, 29setcbas 18061 . . . . . . . . 9 (πœ‘ β†’ π‘ˆ = (Baseβ€˜π‘†))
4240, 41eleqtrrd 2828 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) ∈ π‘ˆ)
4342adantr 479 . . . . . . 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 8040 . . . . . . . . . . . 12 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜πΉ)(𝑂 Func 𝑆)(2nd β€˜πΉ))
4532, 1, 44sylancr 585 . . . . . . . . . . 11 (πœ‘ β†’ (1st β€˜πΉ)(𝑂 Func 𝑆)(2nd β€˜πΉ))
4621, 31, 45funcf1 17846 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π‘†))
4746, 2ffvelcdmd 7088 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜πΉ)β€˜π‘‹) ∈ (Baseβ€˜π‘†))
4847, 41eleqtrrd 2828 . . . . . . . 8 (πœ‘ β†’ ((1st β€˜πΉ)β€˜π‘‹) ∈ π‘ˆ)
4948adantr 479 . . . . . . 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 18064 . . . . . 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 2725 . . . . . . . 8 (Hom β€˜πΆ) = (Hom β€˜πΆ)
53 yoneda.1 . . . . . . . 8 1 = (Idβ€˜πΆ)
5420, 52, 53, 34, 2catidcl 17656 . . . . . . 7 (πœ‘ β†’ ( 1 β€˜π‘‹) ∈ (𝑋(Hom β€˜πΆ)𝑋))
5533, 20, 34, 2, 52, 2yon11 18250 . . . . . . 7 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) = (𝑋(Hom β€˜πΆ)𝑋))
5654, 55eleqtrrd 2828 . . . . . 6 (πœ‘ β†’ ( 1 β€˜π‘‹) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹))
5756adantr 479 . . . . 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 7088 . . . 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 7118 . . 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 18259 . . . 4 (πœ‘ β†’ (𝐹(1st β€˜π‘)𝑋) = (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹))
6719oppccat 17698 . . . . . 6 (𝐢 ∈ Cat β†’ 𝑂 ∈ Cat)
6834, 67syl 17 . . . . 5 (πœ‘ β†’ 𝑂 ∈ Cat)
6925setccat 18068 . . . . . 6 (π‘ˆ ∈ V β†’ 𝑆 ∈ Cat)
7029, 69syl 17 . . . . 5 (πœ‘ β†’ 𝑆 ∈ Cat)
7164, 68, 70, 21, 1, 2evlf1 18206 . . . 4 (πœ‘ β†’ (𝐹(1st β€˜πΈ)𝑋) = ((1st β€˜πΉ)β€˜π‘‹))
7215, 66, 71feq123d 6706 . . 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 256 . 2 (πœ‘ β†’ (𝐹𝑀𝑋):(𝐹(1st β€˜π‘)𝑋)⟢(𝐹(1st β€˜πΈ)𝑋))
7415, 73jca 510 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   βˆͺ cun 3939   βŠ† wss 3941  βŸ¨cop 4631   class class class wbr 5144   ↦ cmpt 5227  ran crn 5674  Rel wrel 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7413   ∈ cmpo 7415  1st c1st 7985  2nd c2nd 7986  tpos ctpos 8224  Basecbs 17174  Hom chom 17238  Catccat 17638  Idccid 17639  Homf chomf 17640  oppCatcoppc 17685   Func cfunc 17834   ∘func ccofu 17836   Nat cnat 17925   FuncCat cfuc 17926  SetCatcsetc 18058   Γ—c cxpc 18153   1stF c1stf 18154   2ndF c2ndf 18155   ⟨,⟩F cprf 18156   evalF cevlf 18195  HomFchof 18234  Yoncyon 18235
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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-hom 17251  df-cco 17252  df-cat 17642  df-cid 17643  df-homf 17644  df-comf 17645  df-oppc 17686  df-func 17838  df-cofu 17840  df-nat 17927  df-fuc 17928  df-setc 18059  df-xpc 18157  df-1stf 18158  df-2ndf 18159  df-prf 18160  df-evlf 18199  df-curf 18200  df-hof 18236  df-yon 18237
This theorem is referenced by:  yonedalem3b  18265  yonedalem3  18266  yonedainv  18267
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