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Theorem equivestrcsetc 18184

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 18182	. 2 ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	fullestrcsetc 18183	. 2 ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)
10	2, 5	setcbas 18111	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
11	4, 10	eqtr4id 2816	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = 𝑈)
12	11	eleq2d 2848	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝐶 ↔ 𝑏 ∈ 𝑈))
13		eqid 2762	. . . . . . . . 9 ⊢ {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
14		equivestrcsetc.i	. . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
15	13, 5, 14	1strwunbndx 17261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑈) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
16	15	ex 416	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝑈 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
17	12, 16	sylbid 242	. . . . . 6 ⊢ (𝜑 → (𝑏 ∈ 𝐶 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
18	17	imp 410	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
19	1, 5	estrcbas 18157	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
20	19	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑈 = (Base‘𝐸))
21	3, 20	eqtr4id 2816	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝐵 = 𝑈)
22	18, 21	eleqtrrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵)
23		fveq2 6867	. . . . . . 7 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝐹‘𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
24	23	f1oeq3d 6803	. . . . . 6 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝑖:𝑏–1-1-onto→(𝐹‘𝑎) ↔ 𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
25	24	exbidv 1941	. . . . 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 485	. . . 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 6845	. . . . . 6 ⊢ ( I ↾ 𝑏):𝑏–1-1-onto→𝑏
28	1, 2, 3, 4, 5, 6	funcestrcsetclem1 18172	. . . . . . . . 9 ⊢ ((𝜑 ∧ {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29	22, 28	syldan 600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
30	13	1strbas 17260	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐶 → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
31	30	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
32	29, 31	eqtr4d 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = 𝑏)
33	32	f1oeq3d 6803	. . . . . 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 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
35		resiexg 7893	. . . . . . 7 ⊢ (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V)
36	35	elv 3459	. . . . . 6 ⊢ ( I ↾ 𝑏) ∈ V
37		f1oeq1 6794	. . . . . 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 3565	. . . . 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 3583	. . 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 1141	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 208 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∃wex 1799 ∈ wcel 2142 ∀wral 3076 ∃wrex 3086 Vcvv 3454 {csn 4582 ⟨cop 4588 class class class wbr 5100 ↦ cmpt 5181 I cid 5541 ↾ cres 5649 –1-1-onto→wf1o 6520 ‘cfv 6521 (class class class)co 7396 ∈ cmpo 7398 ↑_m cmap 8808 WUnicwun 10658 ndxcnx 17229 Basecbs 17245 Full cful 17937 Faith cfth 17938 SetCatcsetc 18108 ExtStrCatcestrc 18154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-wun 10660 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-fz 13513 df-struct 17183 df-slot 17218 df-ndx 17230 df-base 17246 df-hom 17310 df-cco 17311 df-cat 17700 df-cid 17701 df-func 17891 df-full 17939 df-fth 17940 df-setc 18109 df-estrc 18155

This theorem is referenced by: (None)

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