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Theorem issubc3 17900
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 18830, 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 17891 . . 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 17895 . . . 4 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → ( 1𝑥) ∈ (𝑥𝐽𝑥))
109ralrimiva 3144 . . 3 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
11 issubc3.1 . . . 4 𝐷 = (𝐶cat 𝐽)
1211, 1subccat 17899 . . 3 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → 𝐷 ∈ Cat)
133, 10, 123jca 1127 . 2 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
14 simpr1 1193 . . 3 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽cat 𝐻)
15 simpr2 1194 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
16 eqid 2735 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
17 eqid 2735 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
18 eqid 2735 . . . . . . . . . 10 (comp‘𝐷) = (comp‘𝐷)
19 simplrr 778 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐷 ∈ Cat)
20 simprl1 1217 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥𝑆)
21 eqid 2735 . . . . . . . . . . . 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 17738 . . . . . . . . . . . . . 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 17869 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 ⊆ (Base‘𝐶))
2911, 21, 23, 24, 28rescbas 17877 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 = (Base‘𝐷))
3020, 29eleqtrd 2841 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ (Base‘𝐷))
31 simprl2 1218 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦𝑆)
3231, 29eleqtrd 2841 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ (Base‘𝐷))
33 simprl3 1219 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑧𝑆)
3433, 29eleqtrd 2841 . . . . . . . . . 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 17879 . . . . . . . . . . . 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 2841 . . . . . . . . . 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 2841 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
4216, 17, 18, 19, 30, 32, 34, 38, 41catcocl 17730 . . . . . . . . 9 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧))
43 eqid 2735 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
4411, 21, 23, 24, 28, 43rescco 17881 . . . . . . . . . . 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 2854 . . . . . . . 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 3200 . . . . . 6 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
5150ralrimivvva 3203 . . . . 5 ((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
52513adantr2 1169 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
53 r19.26 3109 . . . 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 17887 . . 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 1086   = wceq 1537  wcel 2106  wral 3059  cop 4637   class class class wbr 5148   × cxp 5687   Fn wfn 6558  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710  Homf chomf 17711  cat cssc 17855  cat cresc 17856  Subcatcsubc 17857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-homf 17715  df-ssc 17858  df-resc 17859  df-subc 17860
This theorem is referenced by:  subsubc  17904
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