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Theorem equivestrcsetc 18326

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 18324	. 2 ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	fullestrcsetc 18325	. 2 ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)
10	2, 5	setcbas 18253	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
11	4, 10	eqtr4id 2814	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = 𝑈)
12	11	eleq2d 2846	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝐶 ↔ 𝑏 ∈ 𝑈))
13		eqid 2760	. . . . . . . . 9 ⊢ {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
14		equivestrcsetc.i	. . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
15	13, 5, 14	1strwunbndx 17403	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑈) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
16	15	ex 418	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝑈 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
17	12, 16	sylbid 243	. . . . . 6 ⊢ (𝜑 → (𝑏 ∈ 𝐶 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
18	17	imp 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
19	1, 5	estrcbas 18299	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
20	19	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑈 = (Base‘𝐸))
21	3, 20	eqtr4id 2814	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝐵 = 𝑈)
22	18, 21	eleqtrrd 2863	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵)
23		fveq2 6885	. . . . . . 7 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝐹‘𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
24	23	f1oeq3d 6821	. . . . . 6 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝑖:𝑏–1-1-onto→(𝐹‘𝑎) ↔ 𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
25	24	exbidv 1953	. . . . 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 487	. . . 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 6863	. . . . . 6 ⊢ ( I ↾ 𝑏):𝑏–1-1-onto→𝑏
28	1, 2, 3, 4, 5, 6	funcestrcsetclem1 18314	. . . . . . . . 9 ⊢ ((𝜑 ∧ {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29	22, 28	syldan 603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
30	13	1strbas 17402	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐶 → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
31	30	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
32	29, 31	eqtr4d 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = 𝑏)
33	32	f1oeq3d 6821	. . . . . 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 261	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
35		resiexg 7924	. . . . . . 7 ⊢ (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V)
36	35	elv 3455	. . . . . 6 ⊢ ( I ↾ 𝑏) ∈ V
37		f1oeq1 6812	. . . . . 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 3560	. . . . 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 18	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
40	22, 26, 39	rspcedvd 3578	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))
41	40	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))
42	8, 9, 41	3jca 1146	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∃wex 1811 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 Vcvv 3450 {csn 4583 ⟨cop 4589 class class class wbr 5102 ↦ cmpt 5185 I cid 5544 ↾ cres 5652 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 ↑_m cmap 8847 WUnicwun 10785 ndxcnx 17371 Basecbs 17387 Full cful 18079 Faith cfth 18080 SetCatcsetc 18250 ExtStrCatcestrc 18296

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-wun 10787 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-fz 13640 df-struct 17325 df-slot 17360 df-ndx 17372 df-base 17388 df-hom 17452 df-cco 17453 df-cat 17842 df-cid 17843 df-func 18033 df-full 18081 df-fth 18082 df-setc 18251 df-estrc 18297

This theorem is used by: (None)

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