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Theorem equivestrcsetc 18113

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 18111	. 2 ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	fullestrcsetc 18112	. 2 ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)
10	2, 5	setcbas 18040	. . . . . . . . 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 17195	. . . . . . . 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 18086	. . . . . . 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 6858	. . . . . . 7 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝐹‘𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
24	23	f1oeq3d 6797	. . . . . 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 6838	. . . . . 6 ⊢ ( I ↾ 𝑏):𝑏–1-1-onto→𝑏
28	1, 2, 3, 4, 5, 6	funcestrcsetclem1 18101	. . . . . . . . 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 17194	. . . . . . . . 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 6797	. . . . . 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 7888	. . . . . . 7 ⊢ (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V)
36	35	elv 3452	. . . . . 6 ⊢ ( I ↾ 𝑏) ∈ V
37		f1oeq1 6788	. . . . . 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 3572	. . . . 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 3590	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))
41	40	ralrimiva 3125	. 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 3447 {csn 4589 ⟨cop 4595 class class class wbr 5107 ↦ cmpt 5188 I cid 5532 ↾ cres 5640 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 ↑_m cmap 8799 WUnicwun 10653 ndxcnx 17163 Basecbs 17179 Full cful 17866 Faith cfth 17867 SetCatcsetc 18037 ExtStrCatcestrc 18083

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-wun 10655 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-cat 17629 df-cid 17630 df-func 17820 df-full 17868 df-fth 17869 df-setc 18038 df-estrc 18084

This theorem is referenced by: (None)

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