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Theorem yoniso 18342
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 17913 . . . 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 18155 . . . . . . 7 (𝜑𝐵 = (𝑉 ∩ Cat))
7 inss2 4246 . . . . . . 7 (𝑉 ∩ Cat) ⊆ Cat
86, 7eqsstrdi 4050 . . . . . 6 (𝜑𝐵 ⊆ Cat)
9 yoniso.c . . . . . 6 (𝜑𝐶𝐵)
108, 9sseldd 3996 . . . . 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 18319 . . . 4 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
17 1st2nd 8063 . . . 4 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
181, 16, 17sylancr 587 . . 3 (𝜑𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
192, 11, 12, 13, 10, 14, 15yonffth 18341 . . . . 5 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
2018, 19eqeltrrd 2840 . . . 4 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21 eqid 2735 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
22 yoniso.e . . . . . 6 𝐸 = (𝑄s ran (1st𝑌))
2311oppccat 17769 . . . . . . . 8 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
2410, 23syl 17 . . . . . . 7 (𝜑𝑂 ∈ Cat)
2512setccat 18139 . . . . . . . 8 (𝑈𝑊𝑆 ∈ Cat)
2614, 25syl 17 . . . . . . 7 (𝜑𝑆 ∈ Cat)
2713, 24, 26fuccat 18027 . . . . . 6 (𝜑𝑄 ∈ Cat)
28 fvex 6920 . . . . . . . 8 (1st𝑌) ∈ V
2928rnex 7933 . . . . . . 7 ran (1st𝑌) ∈ V
3029a1i 11 . . . . . 6 (𝜑 → ran (1st𝑌) ∈ V)
3113fucbas 18016 . . . . . . . . 9 (𝑂 Func 𝑆) = (Base‘𝑄)
32 1st2ndbr 8066 . . . . . . . . . 10 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
331, 16, 32sylancr 587 . . . . . . . . 9 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3421, 31, 33funcf1 17917 . . . . . . . 8 (𝜑 → (1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
3534ffnd 6738 . . . . . . 7 (𝜑 → (1st𝑌) Fn (Base‘𝐶))
36 dffn3 6749 . . . . . . 7 ((1st𝑌) Fn (Base‘𝐶) ↔ (1st𝑌):(Base‘𝐶)⟶ran (1st𝑌))
3735, 36sylib 218 . . . . . 6 (𝜑 → (1st𝑌):(Base‘𝐶)⟶ran (1st𝑌))
3821, 22, 27, 30, 37ffthres2c 17994 . . . . 5 (𝜑 → ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ (1st𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd𝑌)))
39 df-br 5149 . . . . 5 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
40 df-br 5149 . . . . 5 ((1st𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
4138, 39, 403bitr3g 313 . . . 4 (𝜑 → (⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))))
4220, 41mpbid 232 . . 3 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
4318, 42eqeltrd 2839 . 2 (𝜑𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
44 fveq2 6907 . . . . . . . . 9 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (1st ‘((1st𝑌)‘𝑥)) = (1st ‘((1st𝑌)‘𝑦)))
4544fveq1d 6909 . . . . . . . 8 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → ((1st ‘((1st𝑌)‘𝑥))‘𝑥) = ((1st ‘((1st𝑌)‘𝑦))‘𝑥))
4645fveq2d 6911 . . . . . . 7 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)))
47 simpl 482 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
4847, 47jca 511 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
49 eleq1w 2822 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
5049anbi2d 630 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
5150anbi2d 630 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
52 2fveq3 6912 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (1st ‘((1st𝑌)‘𝑦)) = (1st ‘((1st𝑌)‘𝑥)))
5352fveq1d 6909 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = ((1st ‘((1st𝑌)‘𝑥))‘𝑥))
5453fveq2d 6911 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)))
55 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 = 𝑥)
5654, 55eqeq12d 2751 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦 ↔ (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥))
5751, 56imbi12d 344 . . . . . . . . . 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)))
5810adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
59 simprr 773 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
60 eqid 2735 . . . . . . . . . . . . 13 (Hom ‘𝐶) = (Hom ‘𝐶)
61 simprl 771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
622, 21, 58, 59, 60, 61yon11 18321 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
6362fveq2d 6911 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = (𝐹‘(𝑥(Hom ‘𝐶)𝑦)))
64 yoniso.1 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘(𝑥(Hom ‘𝐶)𝑦)) = 𝑦)
6563, 64eqtrd 2775 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦)
6657, 65chvarvv 1996 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥)
6748, 66sylan2 593 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥)
6867, 65eqeq12d 2751 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) ↔ 𝑥 = 𝑦))
6946, 68imbitrid 244 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → 𝑥 = 𝑦))
7069ralrimivva 3200 . . . . 5 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → 𝑥 = 𝑦))
71 dff13 7275 . . . . 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)))
7234, 70, 71sylanbrc 583 . . . 4 (𝜑 → (1st𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
73 f1f1orn 6860 . . . 4 ((1st𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆) → (1st𝑌):(Base‘𝐶)–1-1-onto→ran (1st𝑌))
7472, 73syl 17 . . 3 (𝜑 → (1st𝑌):(Base‘𝐶)–1-1-onto→ran (1st𝑌))
7534frnd 6745 . . . . 5 (𝜑 → ran (1st𝑌) ⊆ (𝑂 Func 𝑆))
7622, 31ressbas2 17283 . . . . 5 (ran (1st𝑌) ⊆ (𝑂 Func 𝑆) → ran (1st𝑌) = (Base‘𝐸))
7775, 76syl 17 . . . 4 (𝜑 → ran (1st𝑌) = (Base‘𝐸))
7877f1oeq3d 6846 . . 3 (𝜑 → ((1st𝑌):(Base‘𝐶)–1-1-onto→ran (1st𝑌) ↔ (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸)))
7974, 78mpbid 232 . 2 (𝜑 → (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))
80 eqid 2735 . . 3 (Base‘𝐸) = (Base‘𝐸)
81 yoniso.eb . . 3 (𝜑𝐸𝐵)
82 yoniso.i . . 3 𝐼 = (Iso‘𝐷)
833, 4, 21, 80, 5, 9, 81, 82catciso 18165 . 2 (𝜑 → (𝑌 ∈ (𝐶𝐼𝐸) ↔ (𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)) ∧ (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))))
8443, 79, 83mpbir2and 713 1 (𝜑𝑌 ∈ (𝐶𝐼𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cin 3962  wss 3963  cop 4637   class class class wbr 5148  ran crn 5690  Rel wrel 5694   Fn wfn 6558  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  s cress 17274  Hom chom 17309  Catccat 17709  Homf chomf 17711  oppCatcoppc 17756  Isociso 17794   Func cfunc 17905   Full cful 17956   Faith cfth 17957   FuncCat cfuc 17997  SetCatcsetc 18129  CatCatccatc 18152  Yoncyon 18306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-homf 17715  df-comf 17716  df-oppc 17757  df-sect 17795  df-inv 17796  df-iso 17797  df-ssc 17858  df-resc 17859  df-subc 17860  df-func 17909  df-idfu 17910  df-cofu 17911  df-full 17958  df-fth 17959  df-nat 17998  df-fuc 17999  df-setc 18130  df-catc 18153  df-xpc 18228  df-1stf 18229  df-2ndf 18230  df-prf 18231  df-evlf 18270  df-curf 18271  df-hof 18307  df-yon 18308
This theorem is referenced by: (None)
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