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Theorem issubc3 17564
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 18443, 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 485 . . . 4 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → 𝐽 ∈ (Subcat‘𝐶))
2 issubc3.h . . . 4 𝐻 = (Homf𝐶)
31, 2subcssc 17555 . . 3 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → 𝐽cat 𝐻)
41adantr 481 . . . . 5 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → 𝐽 ∈ (Subcat‘𝐶))
5 issubc3.a . . . . . 6 (𝜑𝐽 Fn (𝑆 × 𝑆))
65ad2antrr 723 . . . . 5 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → 𝐽 Fn (𝑆 × 𝑆))
7 simpr 485 . . . . 5 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → 𝑥𝑆)
8 issubc3.i . . . . 5 1 = (Id‘𝐶)
94, 6, 7, 8subcidcl 17559 . . . 4 (((𝜑𝐽 ∈ (Subcat‘𝐶)) ∧ 𝑥𝑆) → ( 1𝑥) ∈ (𝑥𝐽𝑥))
109ralrimiva 3103 . . 3 ((𝜑𝐽 ∈ (Subcat‘𝐶)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
11 issubc3.1 . . . 4 𝐷 = (𝐶cat 𝐽)
1211, 1subccat 17563 . . 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 2738 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
17 eqid 2738 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
18 eqid 2738 . . . . . . . . . 10 (comp‘𝐷) = (comp‘𝐷)
19 simplrr 775 . . . . . . . . . 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 2738 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
22 issubc3.c . . . . . . . . . . . . 13 (𝜑𝐶 ∈ Cat)
2322ad2antrr 723 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐶 ∈ Cat)
245ad2antrr 723 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 Fn (𝑆 × 𝑆))
252, 21homffn 17402 . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽cat 𝐻)
2824, 26, 27ssc1 17533 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 ⊆ (Base‘𝐶))
2911, 21, 23, 24, 28rescbas 17541 . . . . . . . . . . 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 778 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑥𝐽𝑦))
3611, 21, 23, 24, 28reschom 17543 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 = (Hom ‘𝐷))
3736oveqd 7292 . . . . . . . . . . 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 779 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑦𝐽𝑧))
4036oveqd 7292 . . . . . . . . . . 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 17394 . . . . . . . . 9 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧))
43 eqid 2738 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
4411, 21, 23, 24, 28, 43rescco 17545 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (comp‘𝐶) = (comp‘𝐷))
4544oveqd 7292 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧))
4645oveqd 7292 . . . . . . . . 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 7292 . . . . . . . . 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 468 . . . . . . 7 ((((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
5049ralrimivva 3123 . . . . . 6 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))
5150ralrimivvva 3127 . . . . 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 3095 . . . 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 481 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐶 ∈ Cat)
565adantr 481 . . . 4 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 Fn (𝑆 × 𝑆))
572, 8, 43, 55, 56issubc2 17551 . . 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 710 . 2 ((𝜑 ∧ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)) → 𝐽 ∈ (Subcat‘𝐶))
5913, 58impbida 798 1 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  cop 4567   class class class wbr 5074   × cxp 5587   Fn wfn 6428  cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374  Homf chomf 17375  cat cssc 17519  cat cresc 17520  Subcatcsubc 17521
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-hom 16986  df-cco 16987  df-cat 17377  df-cid 17378  df-homf 17379  df-ssc 17522  df-resc 17523  df-subc 17524
This theorem is referenced by:  subsubc  17568
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