Theorem fullestrcsetc 18196

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 18194	. 2 ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	funcestrcsetclem8 18192	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
10	5	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑈 ∈ WUni)
11		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
12	1, 2, 3, 4, 5, 6	funcestrcsetclem2 18186	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) ∈ 𝑈)
13	12	adantrr 715	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑎) ∈ 𝑈)
14	1, 2, 3, 4, 5, 6	funcestrcsetclem2 18186	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) ∈ 𝑈)
15	14	adantrl 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) ∈ 𝑈)
16	2, 10, 11, 13, 15	elsetchom 18124	. . . . . . 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 18185	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = (Base‘𝑎))
18	17	adantrr 715	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑎) = (Base‘𝑎))
19	1, 2, 3, 4, 5, 6	funcestrcsetclem1 18185	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = (Base‘𝑏))
20	19	adantrl 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = (Base‘𝑏))
21	18, 20	feq23d 6727	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ:(𝐹‘𝑎)⟶(𝐹‘𝑏) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
22	16, 21	bitrd 278	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
23		fvex 6918	. . . . . . . . . . . . 13 ⊢ (Base‘𝑏) ∈ V
24		fvex 6918	. . . . . . . . . . . . 13 ⊢ (Base‘𝑎) ∈ V
25	23, 24	pm3.2i 469	. . . . . . . . . . . 12 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
26		elmapg 8876	. . . . . . . . . . . 12 ⊢ (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
27	25, 26	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
28	27	biimpar 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
29		equequ2 2022	. . . . . . . . . . 11 ⊢ (𝑘 = ℎ → (ℎ = 𝑘 ↔ ℎ = ℎ))
30	29	adantl 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) ∧ 𝑘 = ℎ) → (ℎ = 𝑘 ↔ ℎ = ℎ))
31		eqidd 2730	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ = ℎ)
32	28, 30, 31	rspcedvd 3622	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = 𝑘)
33		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑎) = (Base‘𝑎)
34		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑏) = (Base‘𝑏)
35	1, 2, 3, 4, 5, 6, 7, 33, 34	funcestrcsetclem6 18190	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
36	35	3expa 1115	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
37	36	eqeq2d 2740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → (ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ℎ = 𝑘))
38	37	rexbidva 3170	. . . . . . . . . 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 479	. . . . . . . . 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 256	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = ((𝑎𝐺𝑏)‘𝑘))
41		eqid 2729	. . . . . . . . . . 11 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
42	1, 5	estrcbas 18169	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
43	3, 42	eqtr4id 2788	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 = 𝑈)
44	43	eleq2d 2815	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ 𝑈))
45	44	biimpcd 248	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝐵 → (𝜑 → 𝑎 ∈ 𝑈))
46	45	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑎 ∈ 𝑈))
47	46	impcom 406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝑈)
48	43	eleq2d 2815	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ 𝑈))
49	48	biimpcd 248	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐵 → (𝜑 → 𝑏 ∈ 𝑈))
50	49	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑏 ∈ 𝑈))
51	50	impcom 406	. . . . . . . . . . 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 3320	. . . . . . . . 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 479	. . . . . . . 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 256	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘))
56	55	ex 411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ:(Base‘𝑎)⟶(Base‘𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
57	22, 56	sylbid 239	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘)))
58	57	ralrimiv 3138	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ∀ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘))
59		dffo3 7121	. . . 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 581	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
61	60	ralrimivva 3195	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
62	3, 11, 41	isfull2 17954	. 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 581	1 ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2100  ∀wral 3054  ∃wrex 3063  Vcvv 3472   class class class wbr 5156   ↦ cmpt 5239   I cid 5583   ↾ cres 5688  ⟶wf 6554  –onto→wfo 6556  ‘cfv 6558  (class class class)co 7430   ∈ cmpo 7432   ↑_m cmap 8863  WUnicwun 10750  Basecbs 17234  Hom chom 17298   Func cfunc 17894   Full cful 17945  SetCatcsetc 18118  ExtStrCatcestrc 18166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5293  ax-sep 5307  ax-nul 5314  ax-pow 5373  ax-pr 5437  ax-un 7751  ax-cnex 11221  ax-resscn 11222  ax-1cn 11223  ax-icn 11224  ax-addcl 11225  ax-addrcl 11226  ax-mulcl 11227  ax-mulrcl 11228  ax-mulcom 11229  ax-addass 11230  ax-mulass 11231  ax-distr 11232  ax-i2m1 11233  ax-1ne0 11234  ax-1rid 11235  ax-rnegex 11236  ax-rrecex 11237  ax-cnre 11238  ax-pre-lttri 11239  ax-pre-lttrn 11240  ax-pre-ltadd 11241  ax-pre-mulgt0 11242

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-nel 3040  df-ral 3055  df-rex 3064  df-rmo 3373  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-pss 3979  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-tp 4641  df-op 4643  df-uni 4919  df-iun 5008  df-br 5157  df-opab 5219  df-mpt 5240  df-tr 5274  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5641  df-we 5643  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-pred 6317  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7386  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7885  df-1st 8011  df-2nd 8012  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8742  df-map 8865  df-ixp 8935  df-en 8983  df-dom 8984  df-sdom 8985  df-fin 8986  df-wun 10752  df-pnf 11307  df-mnf 11308  df-xr 11309  df-ltxr 11310  df-le 11311  df-sub 11503  df-neg 11504  df-nn 12275  df-2 12337  df-3 12338  df-4 12339  df-5 12340  df-6 12341  df-7 12342  df-8 12343  df-9 12344  df-n0 12535  df-z 12621  df-dec 12740  df-uz 12885  df-fz 13549  df-struct 17170  df-slot 17205  df-ndx 17217  df-base 17235  df-hom 17311  df-cco 17312  df-cat 17702  df-cid 17703  df-func 17898  df-full 17947  df-setc 18119  df-estrc 18167

This theorem is referenced by:  equivestrcsetc  18197