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Theorem issubc3 16332
 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 17171, 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 16323 . . 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 16327 . . . 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 16331 . . 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 16176 . . . . . . . . . . . . . 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 16304 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝑆 ⊆ (Base‘𝐶))
2911, 21, 23, 24, 28rescbas 16312 . . . . . . . . . . 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 16313 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → 𝐽 = (Hom ‘𝐷))
3736oveqd 6566 . . . . . . . . . . 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 6566 . . . . . . . . . . 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 16169 . . . . . . . . 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 16315 . . . . . . . . . . 11 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (comp‘𝐶) = (comp‘𝐷))
4544oveqd 6566 . . . . . . . . . 10 (((𝜑 ∧ (𝐽cat 𝐻𝐷 ∈ Cat)) ∧ ((𝑥𝑆𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧))
4645oveqd 6566 . . . . . . . . 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 6566 . . . . . . . . 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 16319 . . 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 4583   × cxp 5036   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  Homf chomf 16150   ⊆cat cssc 16290   ↾cat cresc 16291  Subcatcsubc 16292 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 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 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 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-homf 16154  df-ssc 16293  df-resc 16294  df-subc 16295 This theorem is referenced by:  subsubc  16336
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