Theorem fullsetcestrc 17418

Description: The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is full. (Contributed by AV, 1-Apr-2020.)

Hypotheses

Ref	Expression
funcsetcestrc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c	⊢ 𝐶 = (Base‘𝑆)
funcsetcestrc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcsetcestrc.o	⊢ (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))
funcsetcestrc.e	⊢ 𝐸 = (ExtStrCat‘𝑈)

Assertion

Ref	Expression
fullsetcestrc	⊢ (𝜑 → 𝐹(𝑆 Full 𝐸)𝐺)

Distinct variable groups:   𝑥,𝐶   𝜑,𝑥   𝑦,𝐶,𝑥   𝜑,𝑦   𝑥,𝐸

Allowed substitution hints:   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem fullsetcestrc

Dummy variables 𝑎 𝑏 ℎ 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcsetcestrc.s	. . 3 ⊢ 𝑆 = (SetCat‘𝑈)
2		funcsetcestrc.c	. . 3 ⊢ 𝐶 = (Base‘𝑆)
3		funcsetcestrc.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
4		funcsetcestrc.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
5		funcsetcestrc.o	. . 3 ⊢ (𝜑 → ω ∈ 𝑈)
6		funcsetcestrc.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))
7		funcsetcestrc.e	. . 3 ⊢ 𝐸 = (ExtStrCat‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	funcsetcestrc 17416	. 2 ⊢ (𝜑 → 𝐹(𝑆 Func 𝐸)𝐺)
9	1, 2, 3, 4, 5, 6, 7	funcsetcestrclem8 17414	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)))
10	4	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑈 ∈ WUni)
11		eqid 2823	. . . . . . 7 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
12	1, 2, 3, 4, 5	funcsetcestrclem2 17407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → (𝐹‘𝑎) ∈ 𝑈)
13	12	adantrr 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐹‘𝑎) ∈ 𝑈)
14	1, 2, 3, 4, 5	funcsetcestrclem2 17407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘𝑏) ∈ 𝑈)
15	14	adantrl 714	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐹‘𝑏) ∈ 𝑈)
16		eqid 2823	. . . . . . 7 ⊢ (Base‘(𝐹‘𝑎)) = (Base‘(𝐹‘𝑎))
17		eqid 2823	. . . . . . 7 ⊢ (Base‘(𝐹‘𝑏)) = (Base‘(𝐹‘𝑏))
18	7, 10, 11, 13, 15, 16, 17	elestrchom 17380	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)) ↔ ℎ:(Base‘(𝐹‘𝑎))⟶(Base‘(𝐹‘𝑏))))
19	1, 2, 3	funcsetcestrclem1 17406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → (𝐹‘𝑎) = {⟨(Base‘ndx), 𝑎⟩})
20	19	adantrr 715	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐹‘𝑎) = {⟨(Base‘ndx), 𝑎⟩})
21	20	fveq2d 6676	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(𝐹‘𝑎)) = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
22		eqid 2823	. . . . . . . . . . 11 ⊢ {⟨(Base‘ndx), 𝑎⟩} = {⟨(Base‘ndx), 𝑎⟩}
23	22	1strbas 16601	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐶 → 𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
24	23	ad2antrl 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
25	21, 24	eqtr4d 2861	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(𝐹‘𝑎)) = 𝑎)
26	1, 2, 3	funcsetcestrclem1 17406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘𝑏) = {⟨(Base‘ndx), 𝑏⟩})
27	26	adantrl 714	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐹‘𝑏) = {⟨(Base‘ndx), 𝑏⟩})
28	27	fveq2d 6676	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(𝐹‘𝑏)) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29		eqid 2823	. . . . . . . . . . 11 ⊢ {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
30	29	1strbas 16601	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐶 → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
31	30	ad2antll 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
32	28, 31	eqtr4d 2861	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(𝐹‘𝑏)) = 𝑏)
33	25, 32	feq23d 6511	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (ℎ:(Base‘(𝐹‘𝑎))⟶(Base‘(𝐹‘𝑏)) ↔ ℎ:𝑎⟶𝑏))
34		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶))
35	34	ancomd 464	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑏 ∈ 𝐶 ∧ 𝑎 ∈ 𝐶))
36		elmapg 8421	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝐶 ∧ 𝑎 ∈ 𝐶) → (ℎ ∈ (𝑏 ↑m 𝑎) ↔ ℎ:𝑎⟶𝑏))
37	35, 36	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (ℎ ∈ (𝑏 ↑m 𝑎) ↔ ℎ:𝑎⟶𝑏))
38	37	biimpar 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ℎ ∈ (𝑏 ↑m 𝑎))
39		equequ2 2033	. . . . . . . . . . . 12 ⊢ (𝑘 = ℎ → (ℎ = 𝑘 ↔ ℎ = ℎ))
40	39	adantl 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) ∧ 𝑘 = ℎ) → (ℎ = 𝑘 ↔ ℎ = ℎ))
41		eqidd 2824	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ℎ = ℎ)
42	38, 40, 41	rspcedvd 3628	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ∃𝑘 ∈ (𝑏 ↑m 𝑎)ℎ = 𝑘)
43	1, 2, 3, 4, 5, 6	funcsetcestrclem6 17412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) ∧ 𝑘 ∈ (𝑏 ↑m 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
44	43	3expa 1114	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ 𝑘 ∈ (𝑏 ↑m 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
45	44	eqeq2d 2834	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ 𝑘 ∈ (𝑏 ↑m 𝑎)) → (ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ℎ = 𝑘))
46	45	rexbidva 3298	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (∃𝑘 ∈ (𝑏 ↑m 𝑎)ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏 ↑m 𝑎)ℎ = 𝑘))
47	46	adantr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → (∃𝑘 ∈ (𝑏 ↑m 𝑎)ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏 ↑m 𝑎)ℎ = 𝑘))
48	42, 47	mpbird 259	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ∃𝑘 ∈ (𝑏 ↑m 𝑎)ℎ = ((𝑎𝐺𝑏)‘𝑘))
49		eqid 2823	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
50	1, 4	setcbas 17340	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
51	50, 2	syl6reqr 2877	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 = 𝑈)
52	51	eleq2d 2900	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑎 ∈ 𝐶 ↔ 𝑎 ∈ 𝑈))
53	52	biimpcd 251	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝐶 → (𝜑 → 𝑎 ∈ 𝑈))
54	53	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝜑 → 𝑎 ∈ 𝑈))
55	54	impcom 410	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 ∈ 𝑈)
56	51	eleq2d 2900	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑏 ∈ 𝐶 ↔ 𝑏 ∈ 𝑈))
57	56	biimpcd 251	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐶 → (𝜑 → 𝑏 ∈ 𝑈))
58	57	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝜑 → 𝑏 ∈ 𝑈))
59	58	impcom 410	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 ∈ 𝑈)
60	1, 10, 49, 55, 59	setchom 17342	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎(Hom ‘𝑆)𝑏) = (𝑏 ↑m 𝑎))
61	60	rexeqdv 3418	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏 ↑m 𝑎)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
62	61	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → (∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏 ↑m 𝑎)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
63	48, 62	mpbird 259	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘))
64	63	ex 415	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (ℎ:𝑎⟶𝑏 → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
65	33, 64	sylbid 242	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (ℎ:(Base‘(𝐹‘𝑎))⟶(Base‘(𝐹‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
66	18, 65	sylbid 242	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
67	66	ralrimiv 3183	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ∀ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘))
68		dffo3 6870	. . . 4 ⊢ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)) ↔ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)) ∧ ∀ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
69	9, 67, 68	sylanbrc 585	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)))
70	69	ralrimivva 3193	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)))
71	2, 11, 49	isfull2 17183	. 2 ⊢ (𝐹(𝑆 Full 𝐸)𝐺 ↔ (𝐹(𝑆 Func 𝐸)𝐺 ∧ ∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏))))
72	8, 70, 71	sylanbrc 585	1 ⊢ (𝜑 → 𝐹(𝑆 Full 𝐸)𝐺)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  {csn 4569  ⟨cop 4575   class class class wbr 5068   ↦ cmpt 5148   I cid 5461   ↾ cres 5559  ⟶wf 6353  –onto→wfo 6355  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  ωcom 7582   ↑_m cmap 8408  WUnicwun 10124  ndxcnx 16482  Basecbs 16485  Hom chom 16578   Func cfunc 17126   Full cful 17174  SetCatcsetc 17337  ExtStrCatcestrc 17374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-omul 8109  df-er 8291  df-ec 8293  df-qs 8297  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-wun 10126  df-ni 10296  df-pli 10297  df-mi 10298  df-lti 10299  df-plpq 10332  df-mpq 10333  df-ltpq 10334  df-enq 10335  df-nq 10336  df-erq 10337  df-plq 10338  df-mq 10339  df-1nq 10340  df-rq 10341  df-ltnq 10342  df-np 10405  df-plp 10407  df-ltp 10409  df-enr 10479  df-nr 10480  df-c 10545  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-hom 16591  df-cco 16592  df-cat 16941  df-cid 16942  df-func 17130  df-full 17176  df-setc 17338  df-estrc 17375

This theorem is referenced by: (None)