Theorem fullestrcsetc 18195

Description: The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is full. (Contributed by AV, 2-Apr-2020.)

Hypotheses

Ref	Expression
funcestrcsetc.e	⊢ 𝐸 = (ExtStrCat‘𝑈)
funcestrcsetc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcestrcsetc.b	⊢ 𝐵 = (Base‘𝐸)
funcestrcsetc.c	⊢ 𝐶 = (Base‘𝑆)
funcestrcsetc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcestrcsetc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))

Assertion

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

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

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

Proof of Theorem fullestrcsetc

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

Step	Hyp	Ref	Expression
1		funcestrcsetc.e	. . 3 ⊢ 𝐸 = (ExtStrCat‘𝑈)
2		funcestrcsetc.s	. . 3 ⊢ 𝑆 = (SetCat‘𝑈)
3		funcestrcsetc.b	. . 3 ⊢ 𝐵 = (Base‘𝐸)
4		funcestrcsetc.c	. . 3 ⊢ 𝐶 = (Base‘𝑆)
5		funcestrcsetc.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
6		funcestrcsetc.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
7		funcestrcsetc.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
8	1, 2, 3, 4, 5, 6, 7	funcestrcsetc 18193	. 2 ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	funcestrcsetclem8 18191	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
10	5	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑈 ∈ WUni)
11		eqid 2765	. . . . . . . 8 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
12	1, 2, 3, 4, 5, 6	funcestrcsetclem2 18185	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) ∈ 𝑈)
13	12	adantrr 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑎) ∈ 𝑈)
14	1, 2, 3, 4, 5, 6	funcestrcsetclem2 18185	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) ∈ 𝑈)
15	14	adantrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) ∈ 𝑈)
16	2, 10, 11, 13, 15	elsetchom 18126	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)) ↔ ℎ:(𝐹‘𝑎)⟶(𝐹‘𝑏)))
17	1, 2, 3, 4, 5, 6	funcestrcsetclem1 18184	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = (Base‘𝑎))
18	17	adantrr 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑎) = (Base‘𝑎))
19	1, 2, 3, 4, 5, 6	funcestrcsetclem1 18184	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = (Base‘𝑏))
20	19	adantrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = (Base‘𝑏))
21	18, 20	feq23d 6690	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ:(𝐹‘𝑎)⟶(𝐹‘𝑏) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
22	16, 21	bitrd 282	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
23		fvex 6884	. . . . . . . . . . . . 13 ⊢ (Base‘𝑏) ∈ V
24		fvex 6884	. . . . . . . . . . . . 13 ⊢ (Base‘𝑎) ∈ V
25	23, 24	pm3.2i 475	. . . . . . . . . . . 12 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
26		elmapg 8824	. . . . . . . . . . . 12 ⊢ (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
27	25, 26	mp1i 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
28	27	biimpar 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
29		equequ2 2049	. . . . . . . . . . 11 ⊢ (𝑘 = ℎ → (ℎ = 𝑘 ↔ ℎ = ℎ))
30	29	adantl 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) ∧ 𝑘 = ℎ) → (ℎ = 𝑘 ↔ ℎ = ℎ))
31		eqidd 2766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ = ℎ)
32	28, 30, 31	rspcedvd 3586	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = 𝑘)
33		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑎) = (Base‘𝑎)
34		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑏) = (Base‘𝑏)
35	1, 2, 3, 4, 5, 6, 7, 33, 34	funcestrcsetclem6 18189	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
36	35	3expa 1134	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
37	36	eqeq2d 2776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → (ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ℎ = 𝑘))
38	37	rexbidva 3187	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = 𝑘))
39	38	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = 𝑘))
40	32, 39	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = ((𝑎𝐺𝑏)‘𝑘))
41		eqid 2765	. . . . . . . . . . 11 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
42	1, 5	estrcbas 18169	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
43	3, 42	eqtr4id 2819	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 = 𝑈)
44	43	eleq2d 2851	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ 𝑈))
45	44	biimpcd 252	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝐵 → (𝜑 → 𝑎 ∈ 𝑈))
46	45	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑎 ∈ 𝑈))
47	46	impcom 412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝑈)
48	43	eleq2d 2851	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ 𝑈))
49	48	biimpcd 252	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐵 → (𝜑 → 𝑏 ∈ 𝑈))
50	49	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑏 ∈ 𝑈))
51	50	impcom 412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝑈)
52	1, 10, 41, 47, 51, 33, 34	estrchom 18171	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
53	52	rexeqdv 3324	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = ((𝑎𝐺𝑏)‘𝑘)))
54	53	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = ((𝑎𝐺𝑏)‘𝑘)))
55	40, 54	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘))
56	55	ex 417	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ:(Base‘𝑎)⟶(Base‘𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
57	22, 56	sylbid 243	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
58	57	ralrimiv 3156	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ∀ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘))
59		dffo3 7087	. . . 4 ⊢ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)) ↔ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)) ∧ ∀ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
60	9, 58, 59	sylanbrc 594	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
61	60	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
62	3, 11, 41	isfull2 17958	. 2 ⊢ (𝐹(𝐸 Full 𝑆)𝐺 ↔ (𝐹(𝐸 Func 𝑆)𝐺 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏))))
63	8, 61, 62	sylanbrc 594	1 ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  Vcvv 3457   class class class wbr 5104   ↦ cmpt 5185   I cid 5545   ↾ cres 5653  ⟶wf 6521  –onto→wfo 6523  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402   ↑_m cmap 8812  WUnicwun 10673  Basecbs 17257  Hom chom 17309   Func cfunc 17899   Full cful 17949  SetCatcsetc 18120  ExtStrCatcestrc 18166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-wun 10675  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12222  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12493  df-z 12580  df-dec 12700  df-uz 12851  df-fz 13524  df-struct 17195  df-slot 17230  df-ndx 17242  df-base 17258  df-hom 17322  df-cco 17323  df-cat 17712  df-cid 17713  df-func 17903  df-full 17951  df-setc 18121  df-estrc 18167

This theorem is referenced by:  equivestrcsetc  18196