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Theorem equivestrcsetc 18058

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 an equivalence. According to definition 3.33 (1) of [Adamek] p. 36, "A functor F : A -> B is called an equivalence provided that it is full, faithful, and isomorphism-dense in the sense that for any B-object B' there exists some A-object A' such that F(A') is isomorphic to B'.". Therefore, the category of sets and the category of extensible structures are equivalent, according to definition 3.33 (2) of [Adamek] p. 36, "Categories A and B are called equivalent provided that there is an equivalence from A to B.". (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)))))
equivestrcsetc.i	⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)

**Assertion**
Ref	Expression
equivestrcsetc	⊢ (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺 ∧ 𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎)))

Distinct variable groups: 𝑥,𝐵 𝜑,𝑥 𝑥,𝐶 𝑦,𝐵,𝑥 𝜑,𝑦 𝑎,𝑏,𝑥,𝑦,𝐵 𝐹,𝑎,𝑏 𝐺,𝑎,𝑏 𝐸,𝑎,𝑏 𝑆,𝑎,𝑏 𝜑,𝑎,𝑏 𝐶,𝑎 𝑖,𝐹,𝑎,𝑏 Allowed substitution hints: 𝜑(𝑖) 𝐵(𝑖) 𝐶(𝑦,𝑖,𝑏) 𝑆(𝑥,𝑦,𝑖) 𝑈(𝑥,𝑦,𝑖,𝑎,𝑏) 𝐸(𝑥,𝑦,𝑖) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦,𝑖)

**Proof of Theorem equivestrcsetc**
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	fthestrcsetc 18056	. 2 ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	fullestrcsetc 18057	. 2 ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)
10	2, 5	setcbas 17985	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
11	4, 10	eqtr4id 2783	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = 𝑈)
12	11	eleq2d 2814	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝐶 ↔ 𝑏 ∈ 𝑈))
13		eqid 2729	. . . . . . . . 9 ⊢ {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
14		equivestrcsetc.i	. . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
15	13, 5, 14	1strwunbndx 17136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑈) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
16	15	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝑈 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
17	12, 16	sylbid 240	. . . . . 6 ⊢ (𝜑 → (𝑏 ∈ 𝐶 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
18	17	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
19	1, 5	estrcbas 18031	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
20	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑈 = (Base‘𝐸))
21	3, 20	eqtr4id 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝐵 = 𝑈)
22	18, 21	eleqtrrd 2831	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵)
23		fveq2 6822	. . . . . . 7 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝐹‘𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
24	23	f1oeq3d 6761	. . . . . 6 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝑖:𝑏–1-1-onto→(𝐹‘𝑎) ↔ 𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
25	24	exbidv 1921	. . . . 5 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎) ↔ ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
26	25	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑎 = {⟨(Base‘ndx), 𝑏⟩}) → (∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎) ↔ ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
27		f1oi 6802	. . . . . 6 ⊢ ( I ↾ 𝑏):𝑏–1-1-onto→𝑏
28	1, 2, 3, 4, 5, 6	funcestrcsetclem1 18046	. . . . . . . . 9 ⊢ ((𝜑 ∧ {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29	22, 28	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
30	13	1strbas 17135	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐶 → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
31	30	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
32	29, 31	eqtr4d 2767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = 𝑏)
33	32	f1oeq3d 6761	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏–1-1-onto→𝑏))
34	27, 33	mpbiri 258	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
35		resiexg 7845	. . . . . . 7 ⊢ (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V)
36	35	elv 3441	. . . . . 6 ⊢ ( I ↾ 𝑏) ∈ V
37		f1oeq1 6752	. . . . . 6 ⊢ (𝑖 = ( I ↾ 𝑏) → (𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
38	36, 37	spcev 3561	. . . . 5 ⊢ (( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) → ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
39	34, 38	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
40	22, 26, 39	rspcedvd 3579	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))
41	40	ralrimiva 3121	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))
42	8, 9, 41	3jca 1128	1 ⊢ (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺 ∧ 𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 Vcvv 3436 {csn 4577 ⟨cop 4583 class class class wbr 5092 ↦ cmpt 5173 I cid 5513 ↾ cres 5621 –1-1-onto→wf1o 6481 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 ↑_m cmap 8753 WUnicwun 10594 ndxcnx 17104 Basecbs 17120 Full cful 17811 Faith cfth 17812 SetCatcsetc 17982 ExtStrCatcestrc 18028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-wun 10596 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-fz 13411 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-hom 17185 df-cco 17186 df-cat 17574 df-cid 17575 df-func 17765 df-full 17813 df-fth 17814 df-setc 17983 df-estrc 18029

This theorem is referenced by: (None)

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