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Theorem yoniso 18174
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 17748 . . . 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 17987 . . . . . . 7 (𝜑𝐵 = (𝑉 ∩ Cat))
7 inss2 4189 . . . . . . 7 (𝑉 ∩ Cat) ⊆ Cat
86, 7eqsstrdi 3998 . . . . . 6 (𝜑𝐵 ⊆ Cat)
9 yoniso.c . . . . . 6 (𝜑𝐶𝐵)
108, 9sseldd 3945 . . . . 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 18151 . . . 4 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
17 1st2nd 7971 . . . 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 18173 . . . . 5 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
2018, 19eqeltrrd 2839 . . . 4 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21 eqid 2736 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
22 yoniso.e . . . . . 6 𝐸 = (𝑄s ran (1st𝑌))
2311oppccat 17604 . . . . . . . 8 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
2410, 23syl 17 . . . . . . 7 (𝜑𝑂 ∈ Cat)
2512setccat 17971 . . . . . . . 8 (𝑈𝑊𝑆 ∈ Cat)
2614, 25syl 17 . . . . . . 7 (𝜑𝑆 ∈ Cat)
2713, 24, 26fuccat 17859 . . . . . 6 (𝜑𝑄 ∈ Cat)
28 fvex 6855 . . . . . . . 8 (1st𝑌) ∈ V
2928rnex 7849 . . . . . . 7 ran (1st𝑌) ∈ V
3029a1i 11 . . . . . 6 (𝜑 → ran (1st𝑌) ∈ V)
3113fucbas 17848 . . . . . . . . 9 (𝑂 Func 𝑆) = (Base‘𝑄)
32 1st2ndbr 7974 . . . . . . . . . 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 17752 . . . . . . . 8 (𝜑 → (1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
3534ffnd 6669 . . . . . . 7 (𝜑 → (1st𝑌) Fn (Base‘𝐶))
36 dffn3 6681 . . . . . . 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 17827 . . . . 5 (𝜑 → ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ (1st𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd𝑌)))
39 df-br 5106 . . . . 5 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
40 df-br 5106 . . . . 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 2838 . 2 (𝜑𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
44 fveq2 6842 . . . . . . . . 9 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (1st ‘((1st𝑌)‘𝑥)) = (1st ‘((1st𝑌)‘𝑦)))
4544fveq1d 6844 . . . . . . . 8 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → ((1st ‘((1st𝑌)‘𝑥))‘𝑥) = ((1st ‘((1st𝑌)‘𝑦))‘𝑥))
4645fveq2d 6846 . . . . . . 7 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)))
47 simpl 483 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
4847, 47jca 512 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
49 eleq1w 2820 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
5049anbi2d 629 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
5150anbi2d 629 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
52 2fveq3 6847 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (1st ‘((1st𝑌)‘𝑦)) = (1st ‘((1st𝑌)‘𝑥)))
5352fveq1d 6844 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = ((1st ‘((1st𝑌)‘𝑥))‘𝑥))
5453fveq2d 6846 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)))
55 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 = 𝑥)
5654, 55eqeq12d 2752 . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . 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 18153 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
6362fveq2d 6846 . . . . . . . . . . 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 2776 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦)
6657, 65chvarvv 2002 . . . . . . . . 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 2752 . . . . . . 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 3197 . . . . 5 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → 𝑥 = 𝑦))
71 dff13 7202 . . . . 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 6795 . . . 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 6676 . . . . 5 (𝜑 → ran (1st𝑌) ⊆ (𝑂 Func 𝑆))
7622, 31ressbas2 17120 . . . . 5 (ran (1st𝑌) ⊆ (𝑂 Func 𝑆) → ran (1st𝑌) = (Base‘𝐸))
7775, 76syl 17 . . . 4 (𝜑 → ran (1st𝑌) = (Base‘𝐸))
7877f1oeq3d 6781 . . 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 2736 . . 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 17997 . 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 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cin 3909  wss 3910  cop 4592   class class class wbr 5105  ran crn 5634  Rel wrel 5638   Fn wfn 6491  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  s cress 17112  Hom chom 17144  Catccat 17544  Homf chomf 17546  oppCatcoppc 17591  Isociso 17629   Func cfunc 17740   Full cful 17789   Faith cfth 17790   FuncCat cfuc 17829  SetCatcsetc 17961  CatCatccatc 17984  Yoncyon 18138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-homf 17550  df-comf 17551  df-oppc 17592  df-sect 17630  df-inv 17631  df-iso 17632  df-ssc 17693  df-resc 17694  df-subc 17695  df-func 17744  df-idfu 17745  df-cofu 17746  df-full 17791  df-fth 17792  df-nat 17830  df-fuc 17831  df-setc 17962  df-catc 17985  df-xpc 18060  df-1stf 18061  df-2ndf 18062  df-prf 18063  df-evlf 18102  df-curf 18103  df-hof 18139  df-yon 18140
This theorem is referenced by: (None)
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