Theorem rngchomfvalALTV 47520

Description: Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)

Hypotheses

Ref	Expression
rngcbasALTV.c	⊢ 𝐶 = (RngCatALTV‘𝑈)
rngcbasALTV.b	⊢ 𝐵 = (Base‘𝐶)
rngcbasALTV.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngchomfvalALTV.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
rngchomfvalALTV	⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))

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

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rngchomfvalALTV

Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngchomfvalALTV.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
2		rngcbasALTV.c	. . . . 5 ⊢ 𝐶 = (RngCatALTV‘𝑈)
3		rngcbasALTV.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		rngcbasALTV.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	2, 4, 3	rngcbasALTV 47519	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
6		eqidd 2726	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))
7		eqidd 2726	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))))
8	2, 3, 5, 6, 7	rngcvalALTV 47518	. . . 4 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩})
9	8	fveq2d 6900	. . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}))
10	1, 9	eqtrid 2777	. 2 ⊢ (𝜑 → 𝐻 = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}))
11	4	fvexi 6910	. . . 4 ⊢ 𝐵 ∈ V
12	11, 11	mpoex 8084	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)) ∈ V
13		catstr 17967	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩} Struct ⟨1, ;15⟩
14		homid 17412	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
15		snsstp2 4822	. . . 4 ⊢ {⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}
16	13, 14, 15	strfv 17192	. . 3 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)) ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}))
17	12, 16	mp1i 13	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}))
18	10, 17	eqtr4d 2768	1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3461  {ctp 4634  ⟨cop 4636   × cxp 5676   ∘ ccom 5682  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7992  2^nd c2nd 7993  1c1 11146  5c5 12308  ;cdc 12715  ndxcnx 17181  Basecbs 17199  Hom chom 17263  compcco 17264   RngHom crnghm 20402  RngCatALTVcrngcALTV 47516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17135  df-slot 17170  df-ndx 17182  df-base 17200  df-hom 17276  df-cco 17277  df-rngcALTV 47517

This theorem is referenced by:  rngchomALTV  47521  rngccofvalALTV  47523  rngchomffvalALTV  47531