Theorem funcestrcsetc 17260

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, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 23-Mar-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
funcestrcsetc	⊢ (𝜑 → 𝐹(𝐸 Func 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 funcestrcsetc

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

Step	Hyp	Ref	Expression
1		funcestrcsetc.b	. 2 ⊢ 𝐵 = (Base‘𝐸)
2		funcestrcsetc.c	. 2 ⊢ 𝐶 = (Base‘𝑆)
3		eqid 2778	. 2 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
4		eqid 2778	. 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
5		eqid 2778	. 2 ⊢ (Id‘𝐸) = (Id‘𝐸)
6		eqid 2778	. 2 ⊢ (Id‘𝑆) = (Id‘𝑆)
7		eqid 2778	. 2 ⊢ (comp‘𝐸) = (comp‘𝐸)
8		eqid 2778	. 2 ⊢ (comp‘𝑆) = (comp‘𝑆)
9		funcestrcsetc.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
10		funcestrcsetc.e	. . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈)
11	10	estrccat 17244	. . 3 ⊢ (𝑈 ∈ WUni → 𝐸 ∈ Cat)
12	9, 11	syl 17	. 2 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		funcestrcsetc.s	. . . 4 ⊢ 𝑆 = (SetCat‘𝑈)
14	13	setccat 17206	. . 3 ⊢ (𝑈 ∈ WUni → 𝑆 ∈ Cat)
15	9, 14	syl 17	. 2 ⊢ (𝜑 → 𝑆 ∈ Cat)
16		funcestrcsetc.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
17	10, 13, 1, 2, 9, 16	funcestrcsetclem3 17253	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
18		funcestrcsetc.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
19	10, 13, 1, 2, 9, 16, 18	funcestrcsetclem4 17254	. 2 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
20	10, 13, 1, 2, 9, 16, 18	funcestrcsetclem8 17258	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
21	10, 13, 1, 2, 9, 16, 18	funcestrcsetclem7 17257	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎𝐺𝑎)‘((Id‘𝐸)‘𝑎)) = ((Id‘𝑆)‘(𝐹‘𝑎)))
22	10, 13, 1, 2, 9, 16, 18	funcestrcsetclem9 17259	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (ℎ ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑏(Hom ‘𝐸)𝑐))) → ((𝑎𝐺𝑐)‘(𝑘(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐)ℎ)) = (((𝑏𝐺𝑐)‘𝑘)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩(comp‘𝑆)(𝐹‘𝑐))((𝑎𝐺𝑏)‘ℎ)))
23	1, 2, 3, 4, 5, 6, 7, 8, 12, 15, 17, 19, 20, 21, 22	isfuncd 16996	1 ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)

Syntax hints:   → wi 4   = wceq 1507   ∈ wcel 2050   class class class wbr 4930   ↦ cmpt 5009   I cid 5312   ↾ cres 5410  ‘cfv 6190  (class class class)co 6978   ∈ cmpo 6980   ↑_𝑚 cmap 8208  WUnicwun 9922  Basecbs 16342  Hom chom 16435  compcco 16436  Catccat 16796  Idccid 16797   Func cfunc 16985  SetCatcsetc 17196  ExtStrCatcestrc 17233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-addrcl 10398  ax-mulcl 10399  ax-mulrcl 10400  ax-mulcom 10401  ax-addass 10402  ax-mulass 10403  ax-distr 10404  ax-i2m1 10405  ax-1ne0 10406  ax-1rid 10407  ax-rnegex 10408  ax-rrecex 10409  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412  ax-pre-ltadd 10413  ax-pre-mulgt0 10414

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-oadd 7911  df-er 8091  df-map 8210  df-ixp 8262  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312  df-wun 9924  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481  df-le 10482  df-sub 10674  df-neg 10675  df-nn 11442  df-2 11506  df-3 11507  df-4 11508  df-5 11509  df-6 11510  df-7 11511  df-8 11512  df-9 11513  df-n0 11711  df-z 11797  df-dec 11915  df-uz 12062  df-fz 12712  df-struct 16344  df-ndx 16345  df-slot 16346  df-base 16348  df-hom 16448  df-cco 16449  df-cat 16800  df-cid 16801  df-func 16989  df-setc 17197  df-estrc 17234

This theorem is referenced by:  fthestrcsetc  17261  fullestrcsetc  17262  funcrngcsetc  43634  funcrngcsetcALT  43635  funcringcsetc  43671