Theorem funcestrcsetc 18164

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 2761	. 2 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
4		eqid 2761	. 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
5		eqid 2761	. 2 ⊢ (Id‘𝐸) = (Id‘𝐸)
6		eqid 2761	. 2 ⊢ (Id‘𝑆) = (Id‘𝑆)
7		eqid 2761	. 2 ⊢ (comp‘𝐸) = (comp‘𝐸)
8		eqid 2761	. 2 ⊢ (comp‘𝑆) = (comp‘𝑆)
9		funcestrcsetc.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
10		funcestrcsetc.e	. . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈)
11	10	estrccat 18148	. . 3 ⊢ (𝑈 ∈ WUni → 𝐸 ∈ Cat)
12	9, 11	syl 17	. 2 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		funcestrcsetc.s	. . . 4 ⊢ 𝑆 = (SetCat‘𝑈)
14	13	setccat 18101	. . 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 18157	. 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 18158	. 2 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
20	10, 13, 1, 2, 9, 16, 18	funcestrcsetclem8 18162	. 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 18161	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎𝐺𝑎)‘((Id‘𝐸)‘𝑎)) = ((Id‘𝑆)‘(𝐹‘𝑎)))
22	10, 13, 1, 2, 9, 16, 18	funcestrcsetclem9 18163	. 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 17881	1 ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141   class class class wbr 5099   ↦ cmpt 5180   I cid 5539   ↾ cres 5647  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394   ↑_m cmap 8803  WUnicwun 10655  Basecbs 17228  Hom chom 17280  compcco 17281  Catccat 17679  Idccid 17680   Func cfunc 17870  SetCatcsetc 18091  ExtStrCatcestrc 18137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  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 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-ixp 8876  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-wun 10657  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-z 12566  df-dec 12686  df-uz 12837  df-fz 13510  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-hom 17293  df-cco 17294  df-cat 17683  df-cid 17684  df-func 17874  df-setc 18092  df-estrc 18138

This theorem is referenced by:  fthestrcsetc  18165  fullestrcsetc  18166  funcrngcsetc  20669  funcrngcsetcALT  20670  funcringcsetc  20703