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Theorem yoniso 17133
Description: If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from 𝐶 into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)
Hypotheses
Ref Expression
yoniso.y 𝑌 = (Yon‘𝐶)
yoniso.o 𝑂 = (oppCat‘𝐶)
yoniso.s 𝑆 = (SetCat‘𝑈)
yoniso.d 𝐷 = (CatCat‘𝑉)
yoniso.b 𝐵 = (Base‘𝐷)
yoniso.i 𝐼 = (Iso‘𝐷)
yoniso.q 𝑄 = (𝑂 FuncCat 𝑆)
yoniso.e 𝐸 = (𝑄s ran (1st𝑌))
yoniso.v (𝜑𝑉𝑋)
yoniso.c (𝜑𝐶𝐵)
yoniso.u (𝜑𝑈𝑊)
yoniso.h (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoniso.eb (𝜑𝐸𝐵)
yoniso.1 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘(𝑥(Hom ‘𝐶)𝑦)) = 𝑦)
Assertion
Ref Expression
yoniso (𝜑𝑌 ∈ (𝐶𝐼𝐸))
Distinct variable groups:   𝑥,𝑦,𝐶   𝑦,𝐹   𝜑,𝑥,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑄(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥)   𝐼(𝑥,𝑦)   𝑂(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem yoniso
StepHypRef Expression
1 relfunc 16729 . . . 4 Rel (𝐶 Func 𝑄)
2 yoniso.y . . . . 5 𝑌 = (Yon‘𝐶)
3 yoniso.d . . . . . . . 8 𝐷 = (CatCat‘𝑉)
4 yoniso.b . . . . . . . 8 𝐵 = (Base‘𝐷)
5 yoniso.v . . . . . . . 8 (𝜑𝑉𝑋)
63, 4, 5catcbas 16954 . . . . . . 7 (𝜑𝐵 = (𝑉 ∩ Cat))
7 inss2 3982 . . . . . . 7 (𝑉 ∩ Cat) ⊆ Cat
86, 7syl6eqss 3804 . . . . . 6 (𝜑𝐵 ⊆ Cat)
9 yoniso.c . . . . . 6 (𝜑𝐶𝐵)
108, 9sseldd 3753 . . . . 5 (𝜑𝐶 ∈ Cat)
11 yoniso.o . . . . 5 𝑂 = (oppCat‘𝐶)
12 yoniso.s . . . . 5 𝑆 = (SetCat‘𝑈)
13 yoniso.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
14 yoniso.u . . . . 5 (𝜑𝑈𝑊)
15 yoniso.h . . . . 5 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
162, 10, 11, 12, 13, 14, 15yoncl 17110 . . . 4 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
17 1st2nd 7363 . . . 4 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
181, 16, 17sylancr 575 . . 3 (𝜑𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
192, 11, 12, 13, 10, 14, 15yonffth 17132 . . . . 5 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
2018, 19eqeltrrd 2851 . . . 4 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21 eqid 2771 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
22 yoniso.e . . . . . 6 𝐸 = (𝑄s ran (1st𝑌))
2311oppccat 16589 . . . . . . . 8 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
2410, 23syl 17 . . . . . . 7 (𝜑𝑂 ∈ Cat)
2512setccat 16942 . . . . . . . 8 (𝑈𝑊𝑆 ∈ Cat)
2614, 25syl 17 . . . . . . 7 (𝜑𝑆 ∈ Cat)
2713, 24, 26fuccat 16837 . . . . . 6 (𝜑𝑄 ∈ Cat)
28 fvex 6342 . . . . . . . 8 (1st𝑌) ∈ V
2928rnex 7247 . . . . . . 7 ran (1st𝑌) ∈ V
3029a1i 11 . . . . . 6 (𝜑 → ran (1st𝑌) ∈ V)
3113fucbas 16827 . . . . . . . . 9 (𝑂 Func 𝑆) = (Base‘𝑄)
32 1st2ndbr 7366 . . . . . . . . . 10 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
331, 16, 32sylancr 575 . . . . . . . . 9 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3421, 31, 33funcf1 16733 . . . . . . . 8 (𝜑 → (1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
35 ffn 6185 . . . . . . . 8 ((1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆) → (1st𝑌) Fn (Base‘𝐶))
3634, 35syl 17 . . . . . . 7 (𝜑 → (1st𝑌) Fn (Base‘𝐶))
37 dffn3 6194 . . . . . . 7 ((1st𝑌) Fn (Base‘𝐶) ↔ (1st𝑌):(Base‘𝐶)⟶ran (1st𝑌))
3836, 37sylib 208 . . . . . 6 (𝜑 → (1st𝑌):(Base‘𝐶)⟶ran (1st𝑌))
3921, 22, 27, 30, 38ffthres2c 16807 . . . . 5 (𝜑 → ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ (1st𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd𝑌)))
40 df-br 4787 . . . . 5 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
41 df-br 4787 . . . . 5 ((1st𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
4239, 40, 413bitr3g 302 . . . 4 (𝜑 → (⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))))
4320, 42mpbid 222 . . 3 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
4418, 43eqeltrd 2850 . 2 (𝜑𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
45 fveq2 6332 . . . . . . . . 9 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (1st ‘((1st𝑌)‘𝑥)) = (1st ‘((1st𝑌)‘𝑦)))
4645fveq1d 6334 . . . . . . . 8 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → ((1st ‘((1st𝑌)‘𝑥))‘𝑥) = ((1st ‘((1st𝑌)‘𝑦))‘𝑥))
4746fveq2d 6336 . . . . . . 7 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)))
48 simpl 468 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
4948, 48jca 501 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
50 eleq1w 2833 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
5150anbi2d 614 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
5251anbi2d 614 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
53 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → ((1st𝑌)‘𝑦) = ((1st𝑌)‘𝑥))
5453fveq2d 6336 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (1st ‘((1st𝑌)‘𝑦)) = (1st ‘((1st𝑌)‘𝑥)))
5554fveq1d 6334 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = ((1st ‘((1st𝑌)‘𝑥))‘𝑥))
5655fveq2d 6336 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)))
57 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 = 𝑥)
5856, 57eqeq12d 2786 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦 ↔ (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥))
5952, 58imbi12d 333 . . . . . . . . . 10 (𝑦 = 𝑥 → (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦) ↔ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥)))
6010adantr 466 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
61 simprr 756 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
62 eqid 2771 . . . . . . . . . . . . 13 (Hom ‘𝐶) = (Hom ‘𝐶)
63 simprl 754 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
642, 21, 60, 61, 62, 63yon11 17112 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
6564fveq2d 6336 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = (𝐹‘(𝑥(Hom ‘𝐶)𝑦)))
66 yoniso.1 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘(𝑥(Hom ‘𝐶)𝑦)) = 𝑦)
6765, 66eqtrd 2805 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦)
6859, 67chvarv 2425 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥)
6949, 68sylan2 580 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥)
7069, 67eqeq12d 2786 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) ↔ 𝑥 = 𝑦))
7147, 70syl5ib 234 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → 𝑥 = 𝑦))
7271ralrimivva 3120 . . . . 5 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → 𝑥 = 𝑦))
73 dff13 6655 . . . . 5 ((1st𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆) ↔ ((1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → 𝑥 = 𝑦)))
7434, 72, 73sylanbrc 572 . . . 4 (𝜑 → (1st𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
75 f1f1orn 6289 . . . 4 ((1st𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆) → (1st𝑌):(Base‘𝐶)–1-1-onto→ran (1st𝑌))
7674, 75syl 17 . . 3 (𝜑 → (1st𝑌):(Base‘𝐶)–1-1-onto→ran (1st𝑌))
77 frn 6193 . . . . . 6 ((1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆) → ran (1st𝑌) ⊆ (𝑂 Func 𝑆))
7834, 77syl 17 . . . . 5 (𝜑 → ran (1st𝑌) ⊆ (𝑂 Func 𝑆))
7922, 31ressbas2 16138 . . . . 5 (ran (1st𝑌) ⊆ (𝑂 Func 𝑆) → ran (1st𝑌) = (Base‘𝐸))
8078, 79syl 17 . . . 4 (𝜑 → ran (1st𝑌) = (Base‘𝐸))
81 f1oeq3 6270 . . . 4 (ran (1st𝑌) = (Base‘𝐸) → ((1st𝑌):(Base‘𝐶)–1-1-onto→ran (1st𝑌) ↔ (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸)))
8280, 81syl 17 . . 3 (𝜑 → ((1st𝑌):(Base‘𝐶)–1-1-onto→ran (1st𝑌) ↔ (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸)))
8376, 82mpbid 222 . 2 (𝜑 → (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))
84 eqid 2771 . . 3 (Base‘𝐸) = (Base‘𝐸)
85 yoniso.eb . . 3 (𝜑𝐸𝐵)
86 yoniso.i . . 3 𝐼 = (Iso‘𝐷)
873, 4, 21, 84, 5, 9, 85, 86catciso 16964 . 2 (𝜑 → (𝑌 ∈ (𝐶𝐼𝐸) ↔ (𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)) ∧ (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))))
8844, 83, 87mpbir2and 692 1 (𝜑𝑌 ∈ (𝐶𝐼𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cin 3722  wss 3723  cop 4322   class class class wbr 4786  ran crn 5250  Rel wrel 5254   Fn wfn 6026  wf 6027  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  Basecbs 16064  s cress 16065  Hom chom 16160  Catccat 16532  Homf chomf 16534  oppCatcoppc 16578  Isociso 16613   Func cfunc 16721   Full cful 16769   Faith cfth 16770   FuncCat cfuc 16809  SetCatcsetc 16932  CatCatccatc 16951  Yoncyon 17097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-tpos 7504  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-hom 16174  df-cco 16175  df-cat 16536  df-cid 16537  df-homf 16538  df-comf 16539  df-oppc 16579  df-sect 16614  df-inv 16615  df-iso 16616  df-ssc 16677  df-resc 16678  df-subc 16679  df-func 16725  df-idfu 16726  df-cofu 16727  df-full 16771  df-fth 16772  df-nat 16810  df-fuc 16811  df-setc 16933  df-catc 16952  df-xpc 17020  df-1stf 17021  df-2ndf 17022  df-prf 17023  df-evlf 17061  df-curf 17062  df-hof 17098  df-yon 17099
This theorem is referenced by: (None)
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