MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  yoniso Structured version   Visualization version   GIF version

Theorem yoniso 18182
Description: If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from 𝐢 into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)
Hypotheses
Ref Expression
yoniso.y π‘Œ = (Yonβ€˜πΆ)
yoniso.o 𝑂 = (oppCatβ€˜πΆ)
yoniso.s 𝑆 = (SetCatβ€˜π‘ˆ)
yoniso.d 𝐷 = (CatCatβ€˜π‘‰)
yoniso.b 𝐡 = (Baseβ€˜π·)
yoniso.i 𝐼 = (Isoβ€˜π·)
yoniso.q 𝑄 = (𝑂 FuncCat 𝑆)
yoniso.e 𝐸 = (𝑄 β†Ύs ran (1st β€˜π‘Œ))
yoniso.v (πœ‘ β†’ 𝑉 ∈ 𝑋)
yoniso.c (πœ‘ β†’ 𝐢 ∈ 𝐡)
yoniso.u (πœ‘ β†’ π‘ˆ ∈ π‘Š)
yoniso.h (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
yoniso.eb (πœ‘ β†’ 𝐸 ∈ 𝐡)
yoniso.1 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ (πΉβ€˜(π‘₯(Hom β€˜πΆ)𝑦)) = 𝑦)
Assertion
Ref Expression
yoniso (πœ‘ β†’ π‘Œ ∈ (𝐢𝐼𝐸))
Distinct variable groups:   π‘₯,𝑦,𝐢   𝑦,𝐹   πœ‘,π‘₯,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦)   𝐷(π‘₯,𝑦)   𝑄(π‘₯,𝑦)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐸(π‘₯,𝑦)   𝐹(π‘₯)   𝐼(π‘₯,𝑦)   𝑂(π‘₯,𝑦)   𝑉(π‘₯,𝑦)   π‘Š(π‘₯,𝑦)   𝑋(π‘₯,𝑦)

Proof of Theorem yoniso
StepHypRef Expression
1 relfunc 17756 . . . 4 Rel (𝐢 Func 𝑄)
2 yoniso.y . . . . 5 π‘Œ = (Yonβ€˜πΆ)
3 yoniso.d . . . . . . . 8 𝐷 = (CatCatβ€˜π‘‰)
4 yoniso.b . . . . . . . 8 𝐡 = (Baseβ€˜π·)
5 yoniso.v . . . . . . . 8 (πœ‘ β†’ 𝑉 ∈ 𝑋)
63, 4, 5catcbas 17995 . . . . . . 7 (πœ‘ β†’ 𝐡 = (𝑉 ∩ Cat))
7 inss2 4193 . . . . . . 7 (𝑉 ∩ Cat) βŠ† Cat
86, 7eqsstrdi 4002 . . . . . 6 (πœ‘ β†’ 𝐡 βŠ† Cat)
9 yoniso.c . . . . . 6 (πœ‘ β†’ 𝐢 ∈ 𝐡)
108, 9sseldd 3949 . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
11 yoniso.o . . . . 5 𝑂 = (oppCatβ€˜πΆ)
12 yoniso.s . . . . 5 𝑆 = (SetCatβ€˜π‘ˆ)
13 yoniso.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
14 yoniso.u . . . . 5 (πœ‘ β†’ π‘ˆ ∈ π‘Š)
15 yoniso.h . . . . 5 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
162, 10, 11, 12, 13, 14, 15yoncl 18159 . . . 4 (πœ‘ β†’ π‘Œ ∈ (𝐢 Func 𝑄))
17 1st2nd 7975 . . . 4 ((Rel (𝐢 Func 𝑄) ∧ π‘Œ ∈ (𝐢 Func 𝑄)) β†’ π‘Œ = ⟨(1st β€˜π‘Œ), (2nd β€˜π‘Œ)⟩)
181, 16, 17sylancr 588 . . 3 (πœ‘ β†’ π‘Œ = ⟨(1st β€˜π‘Œ), (2nd β€˜π‘Œ)⟩)
192, 11, 12, 13, 10, 14, 15yonffth 18181 . . . . 5 (πœ‘ β†’ π‘Œ ∈ ((𝐢 Full 𝑄) ∩ (𝐢 Faith 𝑄)))
2018, 19eqeltrrd 2835 . . . 4 (πœ‘ β†’ ⟨(1st β€˜π‘Œ), (2nd β€˜π‘Œ)⟩ ∈ ((𝐢 Full 𝑄) ∩ (𝐢 Faith 𝑄)))
21 eqid 2733 . . . . . 6 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
22 yoniso.e . . . . . 6 𝐸 = (𝑄 β†Ύs ran (1st β€˜π‘Œ))
2311oppccat 17612 . . . . . . . 8 (𝐢 ∈ Cat β†’ 𝑂 ∈ Cat)
2410, 23syl 17 . . . . . . 7 (πœ‘ β†’ 𝑂 ∈ Cat)
2512setccat 17979 . . . . . . . 8 (π‘ˆ ∈ π‘Š β†’ 𝑆 ∈ Cat)
2614, 25syl 17 . . . . . . 7 (πœ‘ β†’ 𝑆 ∈ Cat)
2713, 24, 26fuccat 17867 . . . . . 6 (πœ‘ β†’ 𝑄 ∈ Cat)
28 fvex 6859 . . . . . . . 8 (1st β€˜π‘Œ) ∈ V
2928rnex 7853 . . . . . . 7 ran (1st β€˜π‘Œ) ∈ V
3029a1i 11 . . . . . 6 (πœ‘ β†’ ran (1st β€˜π‘Œ) ∈ V)
3113fucbas 17856 . . . . . . . . 9 (𝑂 Func 𝑆) = (Baseβ€˜π‘„)
32 1st2ndbr 7978 . . . . . . . . . 10 ((Rel (𝐢 Func 𝑄) ∧ π‘Œ ∈ (𝐢 Func 𝑄)) β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
331, 16, 32sylancr 588 . . . . . . . . 9 (πœ‘ β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
3421, 31, 33funcf1 17760 . . . . . . . 8 (πœ‘ β†’ (1st β€˜π‘Œ):(Baseβ€˜πΆ)⟢(𝑂 Func 𝑆))
3534ffnd 6673 . . . . . . 7 (πœ‘ β†’ (1st β€˜π‘Œ) Fn (Baseβ€˜πΆ))
36 dffn3 6685 . . . . . . 7 ((1st β€˜π‘Œ) Fn (Baseβ€˜πΆ) ↔ (1st β€˜π‘Œ):(Baseβ€˜πΆ)⟢ran (1st β€˜π‘Œ))
3735, 36sylib 217 . . . . . 6 (πœ‘ β†’ (1st β€˜π‘Œ):(Baseβ€˜πΆ)⟢ran (1st β€˜π‘Œ))
3821, 22, 27, 30, 37ffthres2c 17835 . . . . 5 (πœ‘ β†’ ((1st β€˜π‘Œ)((𝐢 Full 𝑄) ∩ (𝐢 Faith 𝑄))(2nd β€˜π‘Œ) ↔ (1st β€˜π‘Œ)((𝐢 Full 𝐸) ∩ (𝐢 Faith 𝐸))(2nd β€˜π‘Œ)))
39 df-br 5110 . . . . 5 ((1st β€˜π‘Œ)((𝐢 Full 𝑄) ∩ (𝐢 Faith 𝑄))(2nd β€˜π‘Œ) ↔ ⟨(1st β€˜π‘Œ), (2nd β€˜π‘Œ)⟩ ∈ ((𝐢 Full 𝑄) ∩ (𝐢 Faith 𝑄)))
40 df-br 5110 . . . . 5 ((1st β€˜π‘Œ)((𝐢 Full 𝐸) ∩ (𝐢 Faith 𝐸))(2nd β€˜π‘Œ) ↔ ⟨(1st β€˜π‘Œ), (2nd β€˜π‘Œ)⟩ ∈ ((𝐢 Full 𝐸) ∩ (𝐢 Faith 𝐸)))
4138, 39, 403bitr3g 313 . . . 4 (πœ‘ β†’ (⟨(1st β€˜π‘Œ), (2nd β€˜π‘Œ)⟩ ∈ ((𝐢 Full 𝑄) ∩ (𝐢 Faith 𝑄)) ↔ ⟨(1st β€˜π‘Œ), (2nd β€˜π‘Œ)⟩ ∈ ((𝐢 Full 𝐸) ∩ (𝐢 Faith 𝐸))))
4220, 41mpbid 231 . . 3 (πœ‘ β†’ ⟨(1st β€˜π‘Œ), (2nd β€˜π‘Œ)⟩ ∈ ((𝐢 Full 𝐸) ∩ (𝐢 Faith 𝐸)))
4318, 42eqeltrd 2834 . 2 (πœ‘ β†’ π‘Œ ∈ ((𝐢 Full 𝐸) ∩ (𝐢 Faith 𝐸)))
44 fveq2 6846 . . . . . . . . 9 (((1st β€˜π‘Œ)β€˜π‘₯) = ((1st β€˜π‘Œ)β€˜π‘¦) β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘₯)) = (1st β€˜((1st β€˜π‘Œ)β€˜π‘¦)))
4544fveq1d 6848 . . . . . . . 8 (((1st β€˜π‘Œ)β€˜π‘₯) = ((1st β€˜π‘Œ)β€˜π‘¦) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘₯))β€˜π‘₯) = ((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯))
4645fveq2d 6850 . . . . . . 7 (((1st β€˜π‘Œ)β€˜π‘₯) = ((1st β€˜π‘Œ)β€˜π‘¦) β†’ (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘₯))β€˜π‘₯)) = (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯)))
47 simpl 484 . . . . . . . . . 10 ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ π‘₯ ∈ (Baseβ€˜πΆ))
4847, 47jca 513 . . . . . . . . 9 ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ π‘₯ ∈ (Baseβ€˜πΆ)))
49 eleq1w 2817 . . . . . . . . . . . . 13 (𝑦 = π‘₯ β†’ (𝑦 ∈ (Baseβ€˜πΆ) ↔ π‘₯ ∈ (Baseβ€˜πΆ)))
5049anbi2d 630 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ↔ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ π‘₯ ∈ (Baseβ€˜πΆ))))
5150anbi2d 630 . . . . . . . . . . 11 (𝑦 = π‘₯ β†’ ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) ↔ (πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ π‘₯ ∈ (Baseβ€˜πΆ)))))
52 2fveq3 6851 . . . . . . . . . . . . . 14 (𝑦 = π‘₯ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘¦)) = (1st β€˜((1st β€˜π‘Œ)β€˜π‘₯)))
5352fveq1d 6848 . . . . . . . . . . . . 13 (𝑦 = π‘₯ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯) = ((1st β€˜((1st β€˜π‘Œ)β€˜π‘₯))β€˜π‘₯))
5453fveq2d 6850 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯)) = (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘₯))β€˜π‘₯)))
55 id 22 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ 𝑦 = π‘₯)
5654, 55eqeq12d 2749 . . . . . . . . . . 11 (𝑦 = π‘₯ β†’ ((πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯)) = 𝑦 ↔ (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘₯))β€˜π‘₯)) = π‘₯))
5751, 56imbi12d 345 . . . . . . . . . 10 (𝑦 = π‘₯ β†’ (((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯)) = 𝑦) ↔ ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ π‘₯ ∈ (Baseβ€˜πΆ))) β†’ (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘₯))β€˜π‘₯)) = π‘₯)))
5810adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ 𝐢 ∈ Cat)
59 simprr 772 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ 𝑦 ∈ (Baseβ€˜πΆ))
60 eqid 2733 . . . . . . . . . . . . 13 (Hom β€˜πΆ) = (Hom β€˜πΆ)
61 simprl 770 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ π‘₯ ∈ (Baseβ€˜πΆ))
622, 21, 58, 59, 60, 61yon11 18161 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯) = (π‘₯(Hom β€˜πΆ)𝑦))
6362fveq2d 6850 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯)) = (πΉβ€˜(π‘₯(Hom β€˜πΆ)𝑦)))
64 yoniso.1 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ (πΉβ€˜(π‘₯(Hom β€˜πΆ)𝑦)) = 𝑦)
6563, 64eqtrd 2773 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯)) = 𝑦)
6657, 65chvarvv 2003 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ π‘₯ ∈ (Baseβ€˜πΆ))) β†’ (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘₯))β€˜π‘₯)) = π‘₯)
6748, 66sylan2 594 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘₯))β€˜π‘₯)) = π‘₯)
6867, 65eqeq12d 2749 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ ((πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘₯))β€˜π‘₯)) = (πΉβ€˜((1st β€˜((1st β€˜π‘Œ)β€˜π‘¦))β€˜π‘₯)) ↔ π‘₯ = 𝑦))
6946, 68imbitrid 243 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ (((1st β€˜π‘Œ)β€˜π‘₯) = ((1st β€˜π‘Œ)β€˜π‘¦) β†’ π‘₯ = 𝑦))
7069ralrimivva 3194 . . . . 5 (πœ‘ β†’ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)(((1st β€˜π‘Œ)β€˜π‘₯) = ((1st β€˜π‘Œ)β€˜π‘¦) β†’ π‘₯ = 𝑦))
71 dff13 7206 . . . . 5 ((1st β€˜π‘Œ):(Baseβ€˜πΆ)–1-1β†’(𝑂 Func 𝑆) ↔ ((1st β€˜π‘Œ):(Baseβ€˜πΆ)⟢(𝑂 Func 𝑆) ∧ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)(((1st β€˜π‘Œ)β€˜π‘₯) = ((1st β€˜π‘Œ)β€˜π‘¦) β†’ π‘₯ = 𝑦)))
7234, 70, 71sylanbrc 584 . . . 4 (πœ‘ β†’ (1st β€˜π‘Œ):(Baseβ€˜πΆ)–1-1β†’(𝑂 Func 𝑆))
73 f1f1orn 6799 . . . 4 ((1st β€˜π‘Œ):(Baseβ€˜πΆ)–1-1β†’(𝑂 Func 𝑆) β†’ (1st β€˜π‘Œ):(Baseβ€˜πΆ)–1-1-ontoβ†’ran (1st β€˜π‘Œ))
7472, 73syl 17 . . 3 (πœ‘ β†’ (1st β€˜π‘Œ):(Baseβ€˜πΆ)–1-1-ontoβ†’ran (1st β€˜π‘Œ))
7534frnd 6680 . . . . 5 (πœ‘ β†’ ran (1st β€˜π‘Œ) βŠ† (𝑂 Func 𝑆))
7622, 31ressbas2 17128 . . . . 5 (ran (1st β€˜π‘Œ) βŠ† (𝑂 Func 𝑆) β†’ ran (1st β€˜π‘Œ) = (Baseβ€˜πΈ))
7775, 76syl 17 . . . 4 (πœ‘ β†’ ran (1st β€˜π‘Œ) = (Baseβ€˜πΈ))
7877f1oeq3d 6785 . . 3 (πœ‘ β†’ ((1st β€˜π‘Œ):(Baseβ€˜πΆ)–1-1-ontoβ†’ran (1st β€˜π‘Œ) ↔ (1st β€˜π‘Œ):(Baseβ€˜πΆ)–1-1-ontoβ†’(Baseβ€˜πΈ)))
7974, 78mpbid 231 . 2 (πœ‘ β†’ (1st β€˜π‘Œ):(Baseβ€˜πΆ)–1-1-ontoβ†’(Baseβ€˜πΈ))
80 eqid 2733 . . 3 (Baseβ€˜πΈ) = (Baseβ€˜πΈ)
81 yoniso.eb . . 3 (πœ‘ β†’ 𝐸 ∈ 𝐡)
82 yoniso.i . . 3 𝐼 = (Isoβ€˜π·)
833, 4, 21, 80, 5, 9, 81, 82catciso 18005 . 2 (πœ‘ β†’ (π‘Œ ∈ (𝐢𝐼𝐸) ↔ (π‘Œ ∈ ((𝐢 Full 𝐸) ∩ (𝐢 Faith 𝐸)) ∧ (1st β€˜π‘Œ):(Baseβ€˜πΆ)–1-1-ontoβ†’(Baseβ€˜πΈ))))
8443, 79, 83mpbir2and 712 1 (πœ‘ β†’ π‘Œ ∈ (𝐢𝐼𝐸))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3447   ∩ cin 3913   βŠ† wss 3914  βŸ¨cop 4596   class class class wbr 5109  ran crn 5638  Rel wrel 5642   Fn wfn 6495  βŸΆwf 6496  β€“1-1β†’wf1 6497  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  Basecbs 17091   β†Ύs cress 17120  Hom chom 17152  Catccat 17552  Homf chomf 17554  oppCatcoppc 17599  Isociso 17637   Func cfunc 17748   Full cful 17797   Faith cfth 17798   FuncCat cfuc 17837  SetCatcsetc 17969  CatCatccatc 17992  Yoncyon 18146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-tpos 8161  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-pm 8774  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-fz 13434  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-hom 17165  df-cco 17166  df-cat 17556  df-cid 17557  df-homf 17558  df-comf 17559  df-oppc 17600  df-sect 17638  df-inv 17639  df-iso 17640  df-ssc 17701  df-resc 17702  df-subc 17703  df-func 17752  df-idfu 17753  df-cofu 17754  df-full 17799  df-fth 17800  df-nat 17838  df-fuc 17839  df-setc 17970  df-catc 17993  df-xpc 18068  df-1stf 18069  df-2ndf 18070  df-prf 18071  df-evlf 18110  df-curf 18111  df-hof 18147  df-yon 18148
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator