Theorem funcringcsetcALTV2 48772

Description: The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
funcringcsetcALTV2.r	⊢ 𝑅 = (RingCat‘𝑈)
funcringcsetcALTV2.s	⊢ 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV2.b	⊢ 𝐵 = (Base‘𝑅)
funcringcsetcALTV2.c	⊢ 𝐶 = (Base‘𝑆)
funcringcsetcALTV2.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcringcsetcALTV2.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV2.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))

Assertion

Ref	Expression
funcringcsetcALTV2	⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)

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

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

Proof of Theorem funcringcsetcALTV2

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

Step	Hyp	Ref	Expression
1		funcringcsetcALTV2.b	. 2 ⊢ 𝐵 = (Base‘𝑅)
2		funcringcsetcALTV2.c	. 2 ⊢ 𝐶 = (Base‘𝑆)
3		eqid 2737	. 2 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
4		eqid 2737	. 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
5		eqid 2737	. 2 ⊢ (Id‘𝑅) = (Id‘𝑅)
6		eqid 2737	. 2 ⊢ (Id‘𝑆) = (Id‘𝑆)
7		eqid 2737	. 2 ⊢ (comp‘𝑅) = (comp‘𝑅)
8		eqid 2737	. 2 ⊢ (comp‘𝑆) = (comp‘𝑆)
9		funcringcsetcALTV2.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
10		funcringcsetcALTV2.r	. . . 4 ⊢ 𝑅 = (RingCat‘𝑈)
11	10	ringccat 20629	. . 3 ⊢ (𝑈 ∈ WUni → 𝑅 ∈ Cat)
12	9, 11	syl 17	. 2 ⊢ (𝜑 → 𝑅 ∈ Cat)
13		funcringcsetcALTV2.s	. . . 4 ⊢ 𝑆 = (SetCat‘𝑈)
14	13	setccat 18041	. . 3 ⊢ (𝑈 ∈ WUni → 𝑆 ∈ Cat)
15	9, 14	syl 17	. 2 ⊢ (𝜑 → 𝑆 ∈ Cat)
16		funcringcsetcALTV2.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
17	10, 13, 1, 2, 9, 16	funcringcsetcALTV2lem3 48765	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
18		funcringcsetcALTV2.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
19	10, 13, 1, 2, 9, 16, 18	funcringcsetcALTV2lem4 48766	. 2 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
20	10, 13, 1, 2, 9, 16, 18	funcringcsetcALTV2lem8 48770	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑅)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝑆)(𝐹‘𝑏)))
21	10, 13, 1, 2, 9, 16, 18	funcringcsetcALTV2lem7 48769	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎𝐺𝑎)‘((Id‘𝑅)‘𝑎)) = ((Id‘𝑆)‘(𝐹‘𝑎)))
22	10, 13, 1, 2, 9, 16, 18	funcringcsetcALTV2lem9 48771	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (ℎ ∈ (𝑎(Hom ‘𝑅)𝑏) ∧ 𝑘 ∈ (𝑏(Hom ‘𝑅)𝑐))) → ((𝑎𝐺𝑐)‘(𝑘(⟨𝑎, 𝑏⟩(comp‘𝑅)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 17821	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   class class class wbr 5086   ↦ cmpt 5167   I cid 5516   ↾ cres 5624  ‘cfv 6490  (class class class)co 7358   ∈ cmpo 7360  WUnicwun 10612  Basecbs 17168  Hom chom 17220  compcco 17221  Catccat 17619  Idccid 17620   Func cfunc 17810  SetCatcsetc 18031   RingHom crh 20438  RingCatcringc 20611

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 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

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 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-pm 8767  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-wun 10614  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-7 12238  df-8 12239  df-9 12240  df-n0 12427  df-z 12514  df-dec 12634  df-uz 12778  df-fz 13451  df-struct 17106  df-sets 17123  df-slot 17141  df-ndx 17153  df-base 17169  df-ress 17190  df-plusg 17222  df-hom 17233  df-cco 17234  df-0g 17393  df-cat 17623  df-cid 17624  df-homf 17625  df-ssc 17766  df-resc 17767  df-subc 17768  df-func 17814  df-setc 18032  df-estrc 18078  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-mhm 18740  df-grp 18901  df-ghm 19177  df-mgp 20111  df-ur 20152  df-ring 20205  df-rhm 20441  df-ringc 20612

This theorem is referenced by: (None)