Theorem fthestrcsetc 18207

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 faithful. (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)))))

Assertion

Ref	Expression
fthestrcsetc	⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)

Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝜑,𝑦

Allowed substitution hints:   𝐶(𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem fthestrcsetc

Dummy variables 𝑎 𝑏 ℎ 𝑘 are mutually distinct and distinct from all other variables.

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	funcestrcsetc 18206	. 2 ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	funcestrcsetclem8 18204	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
10	5	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑈 ∈ WUni)
11		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
12	1, 5	estrcbas 18182	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
13	3, 12	eqtr4id 2817	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 = 𝑈)
14	13	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ 𝑈))
15	14	biimpcd 252	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝐵 → (𝜑 → 𝑎 ∈ 𝑈))
16	15	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑎 ∈ 𝑈))
17	16	impcom 412	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝑈)
18	13	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ 𝑈))
19	18	biimpcd 252	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐵 → (𝜑 → 𝑏 ∈ 𝑈))
20	19	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑏 ∈ 𝑈))
21	20	impcom 412	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝑈)
22		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝑎) = (Base‘𝑎)
23		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝑏) = (Base‘𝑏)
24	1, 10, 11, 17, 21, 22, 23	estrchom 18184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
25	24	eleq2d 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
26	1, 2, 3, 4, 5, 6, 7, 22, 23	funcestrcsetclem6 18202	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘ℎ) = ℎ)
27	26	3expia 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) → ((𝑎𝐺𝑏)‘ℎ) = ℎ))
28	25, 27	sylbid 243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘ℎ) = ℎ))
29	28	com12 33	. . . . . . . . 9 ⊢ (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝑎𝐺𝑏)‘ℎ) = ℎ))
30	29	adantr 485	. . . . . . . 8 ⊢ ((ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝑎𝐺𝑏)‘ℎ) = ℎ))
31	30	impcom 412	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘ℎ) = ℎ)
32	24	eleq2d 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
33	1, 2, 3, 4, 5, 6, 7, 22, 23	funcestrcsetclem6 18202	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
34	33	3expia 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
35	32, 34	sylbid 243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
36	35	com12 33	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
37	36	adantl 486	. . . . . . . 8 ⊢ ((ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
38	37	impcom 412	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
39	31, 38	eqeq12d 2779	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘ℎ) = ((𝑎𝐺𝑏)‘𝑘) ↔ ℎ = 𝑘))
40	39	biimpd 232	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘ℎ) = ((𝑎𝐺𝑏)‘𝑘) → ℎ = 𝑘))
41	40	ralrimivva 3208	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ∀ℎ ∈ (𝑎(Hom ‘𝐸)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)(((𝑎𝐺𝑏)‘ℎ) = ((𝑎𝐺𝑏)‘𝑘) → ℎ = 𝑘))
42		dff13 7254	. . . 4 ⊢ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)) ↔ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)) ∧ ∀ℎ ∈ (𝑎(Hom ‘𝐸)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)(((𝑎𝐺𝑏)‘ℎ) = ((𝑎𝐺𝑏)‘𝑘) → ℎ = 𝑘)))
43	9, 41, 42	sylanbrc 594	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
44	43	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
45		eqid 2763	. . 3 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
46	3, 11, 45	isfth2 17975	. 2 ⊢ (𝐹(𝐸 Faith 𝑆)𝐺 ↔ (𝐹(𝐸 Func 𝑆)𝐺 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏))))
47	8, 44, 46	sylanbrc 594	1 ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   class class class wbr 5110   ↦ cmpt 5193   I cid 5557   ↾ cres 5665  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8825  WUnicwun 10686  Basecbs 17270  Hom chom 17322   Func cfunc 17912   Faith cfth 17963  SetCatcsetc 18133  ExtStrCatcestrc 18179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-wun 10688  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-struct 17208  df-slot 17243  df-ndx 17255  df-base 17271  df-hom 17335  df-cco 17336  df-cat 17725  df-cid 17726  df-func 17916  df-fth 17965  df-setc 18134  df-estrc 18180

This theorem is referenced by:  equivestrcsetc  18209