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Theorem equivestrcsetc 17402

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 17400	. 2 ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	fullestrcsetc 17401	. 2 ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)
10	2, 5	setcbas 17338	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
11	10, 4	syl6reqr 2875	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = 𝑈)
12	11	eleq2d 2898	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝐶 ↔ 𝑏 ∈ 𝑈))
13		eqid 2821	. . . . . . . . 9 ⊢ {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
14		equivestrcsetc.i	. . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
15	13, 5, 14	1strwunbndx 16600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑈) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
16	15	ex 415	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝑈 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
17	12, 16	sylbid 242	. . . . . 6 ⊢ (𝜑 → (𝑏 ∈ 𝐶 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
18	17	imp 409	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
19	1, 5	estrcbas 17375	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
20	19	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑈 = (Base‘𝐸))
21	20, 3	syl6reqr 2875	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝐵 = 𝑈)
22	18, 21	eleqtrrd 2916	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵)
23		fveq2 6670	. . . . . . 7 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝐹‘𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
24	23	f1oeq3d 6612	. . . . . 6 ⊢ (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝑖:𝑏–1-1-onto→(𝐹‘𝑎) ↔ 𝑖:𝑏–1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
25	24	exbidv 1922	. . . . 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 484	. . . 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 6652	. . . . . 6 ⊢ ( I ↾ 𝑏):𝑏–1-1-onto→𝑏
28	1, 2, 3, 4, 5, 6	funcestrcsetclem1 17390	. . . . . . . . 9 ⊢ ((𝜑 ∧ {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29	22, 28	syldan 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
30	13	1strbas 16599	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐶 → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
31	30	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
32	29, 31	eqtr4d 2859	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = 𝑏)
33	32	f1oeq3d 6612	. . . . . 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 7619	. . . . . . 7 ⊢ (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V)
36	35	elv 3499	. . . . . 6 ⊢ ( I ↾ 𝑏) ∈ V
37		f1oeq1 6604	. . . . . 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 3607	. . . . 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 3626	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))
41	40	ralrimiva 3182	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))
42	8, 9, 41	3jca 1124	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 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3494 {csn 4567 ⟨cop 4573 class class class wbr 5066 ↦ cmpt 5146 I cid 5459 ↾ cres 5557 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ↑_m cmap 8406 WUnicwun 10122 ndxcnx 16480 Basecbs 16483 Full cful 17172 Faith cfth 17173 SetCatcsetc 17335 ExtStrCatcestrc 17372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-wun 10124 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-hom 16589 df-cco 16590 df-cat 16939 df-cid 16940 df-func 17128 df-full 17174 df-fth 17175 df-setc 17336 df-estrc 17373

This theorem is referenced by: (None)

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