Theorem fthestrcsetc 18107

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 18106	. 2 ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)
9	1, 2, 3, 4, 5, 6, 7	funcestrcsetclem8 18104	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
10	5	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑈 ∈ WUni)
11		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
12	1, 5	estrcbas 18082	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
13	3, 12	eqtr4id 2791	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 = 𝑈)
14	13	eleq2d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ 𝑈))
15	14	biimpcd 249	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝐵 → (𝜑 → 𝑎 ∈ 𝑈))
16	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑎 ∈ 𝑈))
17	16	impcom 407	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝑈)
18	13	eleq2d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ 𝑈))
19	18	biimpcd 249	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐵 → (𝜑 → 𝑏 ∈ 𝑈))
20	19	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑏 ∈ 𝑈))
21	20	impcom 407	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝑈)
22		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑎) = (Base‘𝑎)
23		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑏) = (Base‘𝑏)
24	1, 10, 11, 17, 21, 22, 23	estrchom 18084	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
25	24	eleq2d 2823	. . . . . . . . . . 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 18102	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘ℎ) = ℎ)
27	26	3expia 1122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) → ((𝑎𝐺𝑏)‘ℎ) = ℎ))
28	25, 27	sylbid 240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘ℎ) = ℎ))
29	28	com12 32	. . . . . . . . 9 ⊢ (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝑎𝐺𝑏)‘ℎ) = ℎ))
30	29	adantr 480	. . . . . . . 8 ⊢ ((ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝑎𝐺𝑏)‘ℎ) = ℎ))
31	30	impcom 407	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘ℎ) = ℎ)
32	24	eleq2d 2823	. . . . . . . . . . 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 18102	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
34	33	3expia 1122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
35	32, 34	sylbid 240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
36	35	com12 32	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
37	36	adantl 481	. . . . . . . 8 ⊢ ((ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
38	37	impcom 407	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
39	31, 38	eqeq12d 2753	. . . . . 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 229	. . . . 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 3181	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ∀ℎ ∈ (𝑎(Hom ‘𝐸)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)(((𝑎𝐺𝑏)‘ℎ) = ((𝑎𝐺𝑏)‘𝑘) → ℎ = 𝑘))
42		dff13 7202	. . . 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 584	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
44	43	ralrimivva 3181	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
45		eqid 2737	. . 3 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
46	3, 11, 45	isfth2 17875	. 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 584	1 ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086   ↦ cmpt 5167   I cid 5518   ↾ cres 5626  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362   ↑_m cmap 8766  WUnicwun 10614  Basecbs 17170  Hom chom 17222   Func cfunc 17812   Faith cfth 17863  SetCatcsetc 18033  ExtStrCatcestrc 18079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-wun 10616  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-func 17816  df-fth 17865  df-setc 18034  df-estrc 18080

This theorem is referenced by:  equivestrcsetc  18109