MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issubc3 Structured version   Visualization version   GIF version

Theorem issubc3 17894
Description: Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 18817, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
issubc3.h 𝐻 = (Homf𝐶)
issubc3.i 1 = (Id‘𝐶)
issubc3.1 𝐷 = (𝐶cat 𝐽)
issubc3.c (𝜑𝐶 ∈ Cat)
issubc3.a (𝜑𝐽 Fn (𝑆 × 𝑆))
Assertion
Ref Expression
issubc3 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐷   𝑥,𝐻   𝜑,𝑥   𝑥,𝐽   𝑥,𝑆
Allowed substitution hint:   1 (𝑥)

Proof of Theorem issubc3
Dummy variables 𝑓 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . 4 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → 𝐽 ∈ (Subcat‘𝐶))
2 issubc3.h . . . 4 𝐻 = (Homf𝐶)
31, 2subcssc 17885 . . 3 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → 𝐽cat 𝐻)
41adantr 480 . . . . 5 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → 𝐽 ∈ (Subcat‘𝐶))
5 issubc3.a . . . . . 6 (𝜑𝐽 Fn (𝑆 × 𝑆))
65ad2antrr 726 . . . . 5 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → 𝐽 Fn (𝑆 × 𝑆))
7 simpr 484 . . . . 5 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → 𝑥𝑆)
8 issubc3.i . . . . 5 1 = (Id‘𝐶)
94, 6, 7, 8subcidcl 17889 . . . 4 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → ( 1𝑥) ∈ (𝑥𝐽𝑥))
109ralrimiva 3146 . . 3 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
11 issubc3.1 . . . 4 𝐷 = (𝐶cat 𝐽)
1211, 1subccat 17893 . . 3 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → 𝐷 ∈ Cat)
133, 10, 123jca 1129 . 2 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
14 simpr1 1195 . . 3 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽cat 𝐻)
15 simpr2 1196 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
16 eqid 2737 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
17 eqid 2737 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
18 eqid 2737 . . . . . . . . . 10 (comp‘𝐷) = (comp‘𝐷)
19 simplrr 778 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐷 ∈ Cat)
20 simprl1 1219 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥𝑆)
21 eqid 2737 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
22 issubc3.c . . . . . . . . . . . . 13 (𝜑𝐶 ∈ Cat)
2322ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐶 ∈ Cat)
245ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 Fn (𝑆 × 𝑆))
252, 21homffn 17736 . . . . . . . . . . . . . 14 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶))
2625a1i 11 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
27 simplrl 777 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽cat 𝐻)
2824, 26, 27ssc1 17865 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 ⊆ (Base‘𝐶))
2911, 21, 23, 24, 28rescbas 17873 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 = (Base‘𝐷))
3020, 29eleqtrd 2843 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ (Base‘𝐷))
31 simprl2 1220 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦𝑆)
3231, 29eleqtrd 2843 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ (Base‘𝐷))
33 simprl3 1221 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧𝑆)
3433, 29eleqtrd 2843 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ (Base‘𝐷))
35 simprrl 781 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥𝐽𝑦))
3611, 21, 23, 24, 28reschom 17874 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 = (Hom ‘𝐷))
3736oveqd 7448 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑥𝐽𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3835, 37eleqtrd 2843 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
39 simprrr 782 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦𝐽𝑧))
4036oveqd 7448 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑦𝐽𝑧) = (𝑦(Hom ‘𝐷)𝑧))
4139, 40eleqtrd 2843 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
4216, 17, 18, 19, 30, 32, 34, 38, 41catcocl 17728 . . . . . . . . 9 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧))
43 eqid 2737 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
4411, 21, 23, 24, 28, 43rescco 17876 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (comp‘𝐶) = (comp‘𝐷))
4544oveqd 7448 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧))
4645oveqd 7448 . . . . . . . . 9 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
4736oveqd 7448 . . . . . . . . 9 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑥𝐽𝑧) = (𝑥(Hom ‘𝐷)𝑧))
4842, 46, 473eltr4d 2856 . . . . . . . 8 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
4948anassrs 467 . . . . . . 7 ((((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
5049ralrimivva 3202 . . . . . 6 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
5150ralrimivvva 3205 . . . . 5 ((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
52513adantr2 1171 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
53 r19.26 3111 . . . 4 (∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) ↔ (∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
5415, 52, 53sylanbrc 583 . . 3 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
5522adantr 480 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐶 ∈ Cat)
565adantr 480 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 Fn (𝑆 × 𝑆))
572, 8, 43, 55, 56issubc2 17881 . . 3 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
5814, 54, 57mpbir2and 713 . 2 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ∈ (Subcat‘𝐶))
5913, 58impbida 801 1 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cop 4632   class class class wbr 5143   × cxp 5683   Fn wfn 6556  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708  Homf chomf 17709  cat cssc 17851  cat cresc 17852  Subcatcsubc 17853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-homf 17713  df-ssc 17854  df-resc 17855  df-subc 17856
This theorem is referenced by:  subsubc  17898
  Copyright terms: Public domain W3C validator