Theorem fullsetcestrc 16629

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 16627	. 2 ⊢ (𝜑 → 𝐹(𝑆 Func 𝐸)𝐺)
9	1, 2, 3, 4, 5, 6, 7	funcsetcestrclem8 16625	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)))
10	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑈 ∈ WUni)
11		eqid 2610	. . . . . . 7 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
12	1, 2, 3, 4, 5	funcsetcestrclem2 16618	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → (𝐹‘𝑎) ∈ 𝑈)
13	12	adantrr 749	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐹‘𝑎) ∈ 𝑈)
14	1, 2, 3, 4, 5	funcsetcestrclem2 16618	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘𝑏) ∈ 𝑈)
15	14	adantrl 748	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐹‘𝑏) ∈ 𝑈)
16		eqid 2610	. . . . . . 7 ⊢ (Base‘(𝐹‘𝑎)) = (Base‘(𝐹‘𝑎))
17		eqid 2610	. . . . . . 7 ⊢ (Base‘(𝐹‘𝑏)) = (Base‘(𝐹‘𝑏))
18	7, 10, 11, 13, 15, 16, 17	elestrchom 16591	. . . . . 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 16617	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → (𝐹‘𝑎) = {⟨(Base‘ndx), 𝑎⟩})
20	19	adantrr 749	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐹‘𝑎) = {⟨(Base‘ndx), 𝑎⟩})
21	20	fveq2d 6107	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(𝐹‘𝑎)) = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
22		eqid 2610	. . . . . . . . . . 11 ⊢ {⟨(Base‘ndx), 𝑎⟩} = {⟨(Base‘ndx), 𝑎⟩}
23	22	1strbas 15806	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐶 → 𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
24	23	ad2antrl 760	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
25	21, 24	eqtr4d 2647	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(𝐹‘𝑎)) = 𝑎)
26	1, 2, 3	funcsetcestrclem1 16617	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘𝑏) = {⟨(Base‘ndx), 𝑏⟩})
27	26	adantrl 748	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐹‘𝑏) = {⟨(Base‘ndx), 𝑏⟩})
28	27	fveq2d 6107	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(𝐹‘𝑏)) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29		eqid 2610	. . . . . . . . . . 11 ⊢ {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
30	29	1strbas 15806	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐶 → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
31	30	ad2antll 761	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
32	28, 31	eqtr4d 2647	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(𝐹‘𝑏)) = 𝑏)
33	25, 32	feq23d 5953	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (ℎ:(Base‘(𝐹‘𝑎))⟶(Base‘(𝐹‘𝑏)) ↔ ℎ:𝑎⟶𝑏))
34		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶))
35	34	ancomd 466	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑏 ∈ 𝐶 ∧ 𝑎 ∈ 𝐶))
36		elmapg 7757	. . . . . . . . . . . . 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 501	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ℎ ∈ (𝑏 ↑𝑚 𝑎))
39		equequ2 1940	. . . . . . . . . . . 12 ⊢ (𝑘 = ℎ → (ℎ = 𝑘 ↔ ℎ = ℎ))
40	39	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) ∧ 𝑘 = ℎ) → (ℎ = 𝑘 ↔ ℎ = ℎ))
41		eqidd 2611	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ℎ = ℎ)
42	38, 40, 41	rspcedvd 3289	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ∃𝑘 ∈ (𝑏 ↑𝑚 𝑎)ℎ = 𝑘)
43	1, 2, 3, 4, 5, 6	funcsetcestrclem6 16623	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) ∧ 𝑘 ∈ (𝑏 ↑𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
44	43	3expa 1257	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ 𝑘 ∈ (𝑏 ↑𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
45	44	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ 𝑘 ∈ (𝑏 ↑𝑚 𝑎)) → (ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ℎ = 𝑘))
46	45	rexbidva 3031	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (∃𝑘 ∈ (𝑏 ↑𝑚 𝑎)ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏 ↑𝑚 𝑎)ℎ = 𝑘))
47	46	adantr 480	. . . . . . . . . 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 246	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ∃𝑘 ∈ (𝑏 ↑𝑚 𝑎)ℎ = ((𝑎𝐺𝑏)‘𝑘))
49		eqid 2610	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
50	1, 4	setcbas 16551	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
51	50, 2	syl6reqr 2663	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 = 𝑈)
52	51	eleq2d 2673	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑎 ∈ 𝐶 ↔ 𝑎 ∈ 𝑈))
53	52	biimpcd 238	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝐶 → (𝜑 → 𝑎 ∈ 𝑈))
54	53	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝜑 → 𝑎 ∈ 𝑈))
55	54	impcom 445	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 ∈ 𝑈)
56	51	eleq2d 2673	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑏 ∈ 𝐶 ↔ 𝑏 ∈ 𝑈))
57	56	biimpcd 238	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐶 → (𝜑 → 𝑏 ∈ 𝑈))
58	57	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝜑 → 𝑏 ∈ 𝑈))
59	58	impcom 445	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 ∈ 𝑈)
60	1, 10, 49, 55, 59	setchom 16553	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎(Hom ‘𝑆)𝑏) = (𝑏 ↑𝑚 𝑎))
61	60	rexeqdv 3122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏 ↑𝑚 𝑎)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
62	61	adantr 480	. . . . . . . . 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 246	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) ∧ ℎ:𝑎⟶𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘))
64	63	ex 449	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (ℎ:𝑎⟶𝑏 → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
65	33, 64	sylbid 229	. . . . . 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 229	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
67	66	ralrimiv 2948	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ∀ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘))
68		dffo3 6282	. . . 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 695	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)))
70	69	ralrimivva 2954	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏)))
71	2, 11, 49	isfull2 16394	. 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 695	1 ⊢ (𝜑 → 𝐹(𝑆 Full 𝐸)𝐺)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {csn 4125  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   ↾ cres 5040  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   ↑_𝑚 cmap 7744  WUnicwun 9401  ndxcnx 15692  Basecbs 15695  Hom chom 15779   Func cfunc 16337   Full cful 16385  SetCatcsetc 16548  ExtStrCatcestrc 16585

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-inf2 8421  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-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-int 4411  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-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wun 9403  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-plp 9684  df-ltp 9686  df-enr 9756  df-nr 9757  df-c 9821  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-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-func 16341  df-full 16387  df-setc 16549  df-estrc 16586

This theorem is referenced by: (None)