Theorem funcestrcsetc 18131

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 2727	. 2 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
4		eqid 2727	. 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
5		eqid 2727	. 2 ⊢ (Id‘𝐸) = (Id‘𝐸)
6		eqid 2727	. 2 ⊢ (Id‘𝑆) = (Id‘𝑆)
7		eqid 2727	. 2 ⊢ (comp‘𝐸) = (comp‘𝐸)
8		eqid 2727	. 2 ⊢ (comp‘𝑆) = (comp‘𝑆)
9		funcestrcsetc.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
10		funcestrcsetc.e	. . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈)
11	10	estrccat 18114	. . 3 ⊢ (𝑈 ∈ WUni → 𝐸 ∈ Cat)
12	9, 11	syl 17	. 2 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		funcestrcsetc.s	. . . 4 ⊢ 𝑆 = (SetCat‘𝑈)
14	13	setccat 18065	. . 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 18124	. 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 18125	. 2 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
20	10, 13, 1, 2, 9, 16, 18	funcestrcsetclem8 18129	. 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 18128	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎𝐺𝑎)‘((Id‘𝐸)‘𝑎)) = ((Id‘𝑆)‘(𝐹‘𝑎)))
22	10, 13, 1, 2, 9, 16, 18	funcestrcsetclem9 18130	. 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 17842	1 ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   class class class wbr 5142   ↦ cmpt 5225   I cid 5569   ↾ cres 5674  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416   ↑_m cmap 8836  WUnicwun 10715  Basecbs 17171  Hom chom 17235  compcco 17236  Catccat 17635  Idccid 17636   Func cfunc 17831  SetCatcsetc 18055  ExtStrCatcestrc 18103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8838  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-wun 10717  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-struct 17107  df-slot 17142  df-ndx 17154  df-base 17172  df-hom 17248  df-cco 17249  df-cat 17639  df-cid 17640  df-func 17835  df-setc 18056  df-estrc 18104

This theorem is referenced by:  fthestrcsetc  18132  fullestrcsetc  18133  funcrngcsetc  20562  funcrngcsetcALT  20563  funcringcsetc  20596