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Theorem yoniso 16857
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 16454 . . . 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 16679 . . . . . . 7 (𝜑𝐵 = (𝑉 ∩ Cat))
7 inss2 3817 . . . . . . 7 (𝑉 ∩ Cat) ⊆ Cat
86, 7syl6eqss 3639 . . . . . 6 (𝜑𝐵 ⊆ Cat)
9 yoniso.c . . . . . 6 (𝜑𝐶𝐵)
108, 9sseldd 3588 . . . . 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 16834 . . . 4 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
17 1st2nd 7166 . . . 4 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
181, 16, 17sylancr 694 . . 3 (𝜑𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
192, 11, 12, 13, 10, 14, 15yonffth 16856 . . . . 5 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
2018, 19eqeltrrd 2699 . . . 4 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21 eqid 2621 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
22 yoniso.e . . . . . 6 𝐸 = (𝑄s ran (1st𝑌))
2311oppccat 16314 . . . . . . . 8 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
2410, 23syl 17 . . . . . . 7 (𝜑𝑂 ∈ Cat)
2512setccat 16667 . . . . . . . 8 (𝑈𝑊𝑆 ∈ Cat)
2614, 25syl 17 . . . . . . 7 (𝜑𝑆 ∈ Cat)
2713, 24, 26fuccat 16562 . . . . . 6 (𝜑𝑄 ∈ Cat)
28 fvex 6163 . . . . . . . 8 (1st𝑌) ∈ V
2928rnex 7054 . . . . . . 7 ran (1st𝑌) ∈ V
3029a1i 11 . . . . . 6 (𝜑 → ran (1st𝑌) ∈ V)
3113fucbas 16552 . . . . . . . . 9 (𝑂 Func 𝑆) = (Base‘𝑄)
32 1st2ndbr 7169 . . . . . . . . . 10 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
331, 16, 32sylancr 694 . . . . . . . . 9 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3421, 31, 33funcf1 16458 . . . . . . . 8 (𝜑 → (1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
35 ffn 6007 . . . . . . . 8 ((1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆) → (1st𝑌) Fn (Base‘𝐶))
3634, 35syl 17 . . . . . . 7 (𝜑 → (1st𝑌) Fn (Base‘𝐶))
37 dffn3 6016 . . . . . . 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 16532 . . . . 5 (𝜑 → ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ (1st𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd𝑌)))
40 df-br 4619 . . . . 5 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
41 df-br 4619 . . . . 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 2698 . 2 (𝜑𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
45 fveq2 6153 . . . . . . . . 9 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (1st ‘((1st𝑌)‘𝑥)) = (1st ‘((1st𝑌)‘𝑦)))
4645fveq1d 6155 . . . . . . . 8 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → ((1st ‘((1st𝑌)‘𝑥))‘𝑥) = ((1st ‘((1st𝑌)‘𝑦))‘𝑥))
4746fveq2d 6157 . . . . . . 7 (((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)))
48 simpl 473 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
4948, 48jca 554 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
50 eleq1 2686 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
5150anbi2d 739 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
5251anbi2d 739 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
53 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → ((1st𝑌)‘𝑦) = ((1st𝑌)‘𝑥))
5453fveq2d 6157 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (1st ‘((1st𝑌)‘𝑦)) = (1st ‘((1st𝑌)‘𝑥)))
5554fveq1d 6155 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = ((1st ‘((1st𝑌)‘𝑥))‘𝑥))
5655fveq2d 6157 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)))
57 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 = 𝑥)
5856, 57eqeq12d 2636 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦 ↔ (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥))
5952, 58imbi12d 334 . . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
61 simprr 795 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
62 eqid 2621 . . . . . . . . . . . . 13 (Hom ‘𝐶) = (Hom ‘𝐶)
63 simprl 793 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
642, 21, 60, 61, 62, 63yon11 16836 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘((1st𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
6564fveq2d 6157 . . . . . . . . . . 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 2655 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑦))‘𝑥)) = 𝑦)
6859, 67chvarv 2262 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥)
6949, 68sylan2 491 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st𝑌)‘𝑥))‘𝑥)) = 𝑥)
7069, 67eqeq12d 2636 . . . . . . 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 2966 . . . . 5 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st𝑌)‘𝑥) = ((1st𝑌)‘𝑦) → 𝑥 = 𝑦))
73 dff13 6472 . . . . 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 697 . . . 4 (𝜑 → (1st𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
75 f1f1orn 6110 . . . 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 6015 . . . . . 6 ((1st𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆) → ran (1st𝑌) ⊆ (𝑂 Func 𝑆))
7834, 77syl 17 . . . . 5 (𝜑 → ran (1st𝑌) ⊆ (𝑂 Func 𝑆))
7922, 31ressbas2 15863 . . . . 5 (ran (1st𝑌) ⊆ (𝑂 Func 𝑆) → ran (1st𝑌) = (Base‘𝐸))
8078, 79syl 17 . . . 4 (𝜑 → ran (1st𝑌) = (Base‘𝐸))
81 f1oeq3 6091 . . . 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 2621 . . 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 16689 . 2 (𝜑 → (𝑌 ∈ (𝐶𝐼𝐸) ↔ (𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)) ∧ (1st𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))))
8844, 83, 87mpbir2and 956 1 (𝜑𝑌 ∈ (𝐶𝐼𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  cin 3558  wss 3559  cop 4159   class class class wbr 4618  ran crn 5080  Rel wrel 5084   Fn wfn 5847  wf 5848  1-1wf1 5849  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119  Basecbs 15792  s cress 15793  Hom chom 15884  Catccat 16257  Homf chomf 16259  oppCatcoppc 16303  Isociso 16338   Func cfunc 16446   Full cful 16494   Faith cfth 16495   FuncCat cfuc 16534  SetCatcsetc 16657  CatCatccatc 16676  Yoncyon 16821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-tpos 7304  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-hom 15898  df-cco 15899  df-cat 16261  df-cid 16262  df-homf 16263  df-comf 16264  df-oppc 16304  df-sect 16339  df-inv 16340  df-iso 16341  df-ssc 16402  df-resc 16403  df-subc 16404  df-func 16450  df-idfu 16451  df-cofu 16452  df-full 16496  df-fth 16497  df-nat 16535  df-fuc 16536  df-setc 16658  df-catc 16677  df-xpc 16744  df-1stf 16745  df-2ndf 16746  df-prf 16747  df-evlf 16785  df-curf 16786  df-hof 16822  df-yon 16823
This theorem is referenced by: (None)
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