Theorem fullestrcsetc 17868

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 17866	. 2 ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	funcestrcsetclem8 17864	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
10	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑈 ∈ WUni)
11		eqid 2738	. . . . . . . 8 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
12	1, 2, 3, 4, 5, 6	funcestrcsetclem2 17858	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) ∈ 𝑈)
13	12	adantrr 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑎) ∈ 𝑈)
14	1, 2, 3, 4, 5, 6	funcestrcsetclem2 17858	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) ∈ 𝑈)
15	14	adantrl 713	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) ∈ 𝑈)
16	2, 10, 11, 13, 15	elsetchom 17796	. . . . . . 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 17857	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = (Base‘𝑎))
18	17	adantrr 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑎) = (Base‘𝑎))
19	1, 2, 3, 4, 5, 6	funcestrcsetclem1 17857	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = (Base‘𝑏))
20	19	adantrl 713	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = (Base‘𝑏))
21	18, 20	feq23d 6595	. . . . . . 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 6787	. . . . . . . . . . . . 13 ⊢ (Base‘𝑏) ∈ V
24		fvex 6787	. . . . . . . . . . . . 13 ⊢ (Base‘𝑎) ∈ V
25	23, 24	pm3.2i 471	. . . . . . . . . . . 12 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
26		elmapg 8628	. . . . . . . . . . . 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 478	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
29		equequ2 2029	. . . . . . . . . . 11 ⊢ (𝑘 = ℎ → (ℎ = 𝑘 ↔ ℎ = ℎ))
30	29	adantl 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) ∧ 𝑘 = ℎ) → (ℎ = 𝑘 ↔ ℎ = ℎ))
31		eqidd 2739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ = ℎ)
32	28, 30, 31	rspcedvd 3563	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))ℎ = 𝑘)
33		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑎) = (Base‘𝑎)
34		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑏) = (Base‘𝑏)
35	1, 2, 3, 4, 5, 6, 7, 33, 34	funcestrcsetclem6 17862	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
36	35	3expa 1117	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
37	36	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → (ℎ = ((𝑎𝐺𝑏)‘𝑘) ↔ ℎ = 𝑘))
38	37	rexbidva 3225	. . . . . . . . . 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 481	. . . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
42	1, 5	estrcbas 17841	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
43	3, 42	eqtr4id 2797	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 = 𝑈)
44	43	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ 𝑈))
45	44	biimpcd 248	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝐵 → (𝜑 → 𝑎 ∈ 𝑈))
46	45	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑎 ∈ 𝑈))
47	46	impcom 408	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝑈)
48	43	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ 𝑈))
49	48	biimpcd 248	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐵 → (𝜑 → 𝑏 ∈ 𝑈))
50	49	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑏 ∈ 𝑈))
51	50	impcom 408	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝑈)
52	1, 10, 41, 47, 51, 33, 34	estrchom 17843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
53	52	rexeqdv 3349	. . . . . . . . 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 481	. . . . . . . 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 413	. . . . . 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 3102	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ∀ℎ ∈ ((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)ℎ = ((𝑎𝐺𝑏)‘𝑘))
59		dffo3 6978	. . . 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 583	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
61	60	ralrimivva 3123	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
62	3, 11, 41	isfull2 17627	. 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 583	1 ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   class class class wbr 5074   ↦ cmpt 5157   I cid 5488   ↾ cres 5591  ⟶wf 6429  –onto→wfo 6431  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ↑_m cmap 8615  WUnicwun 10456  Basecbs 16912  Hom chom 16973   Func cfunc 17569   Full cful 17618  SetCatcsetc 17790  ExtStrCatcestrc 17838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-wun 10458  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-slot 16883  df-ndx 16895  df-base 16913  df-hom 16986  df-cco 16987  df-cat 17377  df-cid 17378  df-func 17573  df-full 17620  df-setc 17791  df-estrc 17839

This theorem is referenced by:  equivestrcsetc  17869