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Theorem yoniso 18305
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 17876 . . . 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 18118 . . . . . . 7 (𝜑𝐵 = (𝑉 ∩ Cat))
7 inss2 4230 . . . . . . 7 (𝑉 ∩ Cat) ⊆ Cat
86, 7eqsstrdi 4033 . . . . . 6 (𝜑𝐵 ⊆ Cat)
9 yoniso.c . . . . . 6 (𝜑𝐶𝐵)
108, 9sseldd 3979 . . . . 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 18282 . . . 4 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
17 1st2nd 8052 . . . 4 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
181, 16, 17sylancr 585 . . 3 (𝜑𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
192, 11, 12, 13, 10, 14, 15yonffth 18304 . . . . 5 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
2018, 19eqeltrrd 2826 . . . 4 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21 eqid 2725 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
22 yoniso.e . . . . . 6 𝐸 = (𝑄s ran (1st𝑌))
2311oppccat 17732 . . . . . . . 8 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
2410, 23syl 17 . . . . . . 7 (𝜑𝑂 ∈ Cat)
2512setccat 18102 . . . . . . . 8 (𝑈𝑊𝑆 ∈ Cat)
2614, 25syl 17 . . . . . . 7 (𝜑𝑆 ∈ Cat)
2713, 24, 26fuccat 17990 . . . . . 6 (𝜑𝑄 ∈ Cat)
28 fvex 6913 . . . . . . . 8 (1st𝑌) ∈ V
2928rnex 7922 . . . . . . 7 ran (1st𝑌) ∈ V
3029a1i 11 . . . . . 6 (𝜑 → ran (1st𝑌) ∈ V)
3113fucbas 17979 . . . . . . . . 9 (𝑂 Func 𝑆) = (Base‘𝑄)
32 1st2ndbr 8055 . . . . . . . . . 10 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
331, 16, 32sylancr 585 . . . . . . . . 9 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3421, 31, 33funcf1 17880 . . . . . . . 8 (𝜑 → (1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
3534ffnd 6728 . . . . . . 7 (𝜑 → (1st𝑌) Fn (Base‘𝐶))
36 dffn3 6739 . . . . . . 7 ((1st𝑌) Fn (Base‘𝐶) ↔ (1st𝑌):(Base‘𝐶)⟶ran (1st𝑌))
3735, 36sylib 217 . . . . . 6 (𝜑 → (1st𝑌):(Base‘𝐶)⟶ran (1st𝑌))
3821, 22, 27, 30, 37ffthres2c 17957 . . . . 5 (𝜑 → ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ (1st𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd𝑌)))
39 df-br 5153 . . . . 5 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
40 df-br 5153 . . . . 5 ((1st𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
4138, 39, 403bitr3g 312 . . . 4 (𝜑 → (⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))))
4220, 41mpbid 231 . . 3 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
4318, 42eqeltrd 2825 . 2 (𝜑𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
44 fveq2 6900 . . . . . . . . 9 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (1st ‘((1st𝑌)‘𝑥)) = (1st ‘((1st𝑌)‘𝑦)))
4544fveq1d 6902 . . . . . . . 8 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → ((1st ‘((1st𝑌)‘𝑥))‘𝑥) = ((1st ‘((1st𝑌)‘𝑦))‘𝑥))
4645fveq2d 6904 . . . . . . 7 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)))
47 simpl 481 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
4847, 47jca 510 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
49 eleq1w 2808 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
5049anbi2d 628 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
5150anbi2d 628 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
52 2fveq3 6905 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (1st ‘((1st𝑌)‘𝑦)) = (1st ‘((1st𝑌)‘𝑥)))
5352fveq1d 6902 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = ((1st ‘((1st𝑌)‘𝑥))‘𝑥))
5453fveq2d 6904 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)))
55 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 = 𝑥)
5654, 55eqeq12d 2741 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦 ↔ (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥))
5751, 56imbi12d 343 . . . . . . . . . 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 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
59 simprr 771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
60 eqid 2725 . . . . . . . . . . . . 13 (Hom ‘𝐶) = (Hom ‘𝐶)
61 simprl 769 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
622, 21, 58, 59, 60, 61yon11 18284 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
6362fveq2d 6904 . . . . . . . . . . 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 2765 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦)
6657, 65chvarvv 1994 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥)
6748, 66sylan2 591 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥)
6867, 65eqeq12d 2741 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) ↔ 𝑥 = 𝑦))
6946, 68imbitrid 243 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → 𝑥 = 𝑦))
7069ralrimivva 3190 . . . . 5 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → 𝑥 = 𝑦))
71 dff13 7269 . . . . 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 581 . . . 4 (𝜑 → (1st𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
73 f1f1orn 6853 . . . 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 6735 . . . . 5 (𝜑 → ran (1st𝑌) ⊆ (𝑂 Func 𝑆))
7622, 31ressbas2 17246 . . . . 5 (ran (1st𝑌) ⊆ (𝑂 Func 𝑆) → ran (1st𝑌) = (Base‘𝐸))
7775, 76syl 17 . . . 4 (𝜑 → ran (1st𝑌) = (Base‘𝐸))
7877f1oeq3d 6839 . . 3 (𝜑 → ((1st𝑌):(Base‘𝐶)–1-1-onto→ran (1st𝑌) ↔ (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸)))
7974, 78mpbid 231 . 2 (𝜑 → (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))
80 eqid 2725 . . 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 18128 . 2 (𝜑 → (𝑌 ∈ (𝐶𝐼𝐸) ↔ (𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)) ∧ (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))))
8443, 79, 83mpbir2and 711 1 (𝜑𝑌 ∈ (𝐶𝐼𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  cin 3945  wss 3946  cop 4638   class class class wbr 5152  ran crn 5682  Rel wrel 5686   Fn wfn 6548  wf 6549  1-1wf1 6550  1-1-ontowf1o 6552  cfv 6553  (class class class)co 7423  1st c1st 8000  2nd c2nd 8001  Basecbs 17208  s cress 17237  Hom chom 17272  Catccat 17672  Homf chomf 17674  oppCatcoppc 17719  Isociso 17757   Func cfunc 17868   Full cful 17919   Faith cfth 17920   FuncCat cfuc 17960  SetCatcsetc 18092  CatCatccatc 18115  Yoncyon 18269
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 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-cnex 11210  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-addrcl 11215  ax-mulcl 11216  ax-mulrcl 11217  ax-mulcom 11218  ax-addass 11219  ax-mulass 11220  ax-distr 11221  ax-i2m1 11222  ax-1ne0 11223  ax-1rid 11224  ax-rnegex 11225  ax-rrecex 11226  ax-cnre 11227  ax-pre-lttri 11228  ax-pre-lttrn 11229  ax-pre-ltadd 11230  ax-pre-mulgt0 11231
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 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7876  df-1st 8002  df-2nd 8003  df-tpos 8240  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-map 8856  df-pm 8857  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-le 11300  df-sub 11492  df-neg 11493  df-nn 12260  df-2 12322  df-3 12323  df-4 12324  df-5 12325  df-6 12326  df-7 12327  df-8 12328  df-9 12329  df-n0 12520  df-z 12606  df-dec 12725  df-uz 12870  df-fz 13534  df-struct 17144  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-hom 17285  df-cco 17286  df-cat 17676  df-cid 17677  df-homf 17678  df-comf 17679  df-oppc 17720  df-sect 17758  df-inv 17759  df-iso 17760  df-ssc 17821  df-resc 17822  df-subc 17823  df-func 17872  df-idfu 17873  df-cofu 17874  df-full 17921  df-fth 17922  df-nat 17961  df-fuc 17962  df-setc 18093  df-catc 18116  df-xpc 18191  df-1stf 18192  df-2ndf 18193  df-prf 18194  df-evlf 18233  df-curf 18234  df-hof 18270  df-yon 18271
This theorem is referenced by: (None)
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