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

Theorem issubc3 16281
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 17120, 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 476 . . . 4 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → 𝐽 ∈ (Subcat‘𝐶))
2 issubc3.h . . . 4 𝐻 = (Homf𝐶)
31, 2subcssc 16272 . . 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 758 . . . . 5 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → 𝐽 Fn (𝑆 × 𝑆))
7 simpr 476 . . . . 5 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → 𝑥𝑆)
8 issubc3.i . . . . 5 1 = (Id‘𝐶)
94, 6, 7, 8subcidcl 16276 . . . 4 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → ( 1𝑥) ∈ (𝑥𝐽𝑥))
109ralrimiva 2949 . . 3 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
11 issubc3.1 . . . 4 𝐷 = (𝐶cat 𝐽)
1211, 1subccat 16280 . . 3 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → 𝐷 ∈ Cat)
133, 10, 123jca 1235 . 2 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
14 simpr1 1060 . . 3 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽cat 𝐻)
15 simpr2 1061 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
16 eqid 2610 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
17 eqid 2610 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
18 eqid 2610 . . . . . . . . . 10 (comp‘𝐷) = (comp‘𝐷)
19 simplrr 797 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐷 ∈ Cat)
20 simprl1 1099 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥𝑆)
21 eqid 2610 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
22 issubc3.c . . . . . . . . . . . . 13 (𝜑𝐶 ∈ Cat)
2322ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐶 ∈ Cat)
245ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 Fn (𝑆 × 𝑆))
252, 21homffn 16125 . . . . . . . . . . . . . 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 796 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽cat 𝐻)
2824, 26, 27ssc1 16253 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 ⊆ (Base‘𝐶))
2911, 21, 23, 24, 28rescbas 16261 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 = (Base‘𝐷))
3020, 29eleqtrd 2690 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ (Base‘𝐷))
31 simprl2 1100 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦𝑆)
3231, 29eleqtrd 2690 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ (Base‘𝐷))
33 simprl3 1101 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧𝑆)
3433, 29eleqtrd 2690 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ (Base‘𝐷))
35 simprrl 800 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥𝐽𝑦))
3611, 21, 23, 24, 28reschom 16262 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 = (Hom ‘𝐷))
3736oveqd 6544 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑥𝐽𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3835, 37eleqtrd 2690 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
39 simprrr 801 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦𝐽𝑧))
4036oveqd 6544 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑦𝐽𝑧) = (𝑦(Hom ‘𝐷)𝑧))
4139, 40eleqtrd 2690 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
4216, 17, 18, 19, 30, 32, 34, 38, 41catcocl 16118 . . . . . . . . 9 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧))
43 eqid 2610 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
4411, 21, 23, 24, 28, 43rescco 16264 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (comp‘𝐶) = (comp‘𝐷))
4544oveqd 6544 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧))
4645oveqd 6544 . . . . . . . . 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 6544 . . . . . . . . 9 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑥𝐽𝑧) = (𝑥(Hom ‘𝐷)𝑧))
4842, 46, 473eltr4d 2703 . . . . . . . 8 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
4948anassrs 678 . . . . . . 7 ((((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
5049ralrimivva 2954 . . . . . 6 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
5150ralrimivvva 2955 . . . . 5 ((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
52513adantr2 1214 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
53 r19.26 3046 . . . 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 695 . . 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 16268 . . 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 959 . 2 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ∈ (Subcat‘𝐶))
5913, 58impbida 873 1 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  cop 4131   class class class wbr 4578   × cxp 5026   Fn wfn 5785  cfv 5790  (class class class)co 6527  Basecbs 15644  Hom chom 15728  compcco 15729  Catccat 16097  Idccid 16098  Homf chomf 16099  cat cssc 16239  cat cresc 16240  Subcatcsubc 16241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-pm 7725  df-ixp 7773  df-en 7820  df-dom 7821  df-sdom 7822  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-dec 11329  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-hom 15742  df-cco 15743  df-cat 16101  df-cid 16102  df-homf 16103  df-ssc 16242  df-resc 16243  df-subc 16244
This theorem is referenced by:  subsubc  16285
  Copyright terms: Public domain W3C validator