Theorem ringchomfvalALTV 44338

Description: Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ringcbasALTV.c	⊢ 𝐶 = (RingCatALTV‘𝑈)
ringcbasALTV.b	⊢ 𝐵 = (Base‘𝐶)
ringcbasALTV.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
ringchomfvalALTV.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
ringchomfvalALTV	⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 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 ringchomfvalALTV

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

Step	Hyp	Ref	Expression
1		ringchomfvalALTV.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
2		ringcbasALTV.c	. . . . 5 ⊢ 𝐶 = (RingCatALTV‘𝑈)
3		ringcbasALTV.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		ringcbasALTV.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	2, 4, 3	ringcbasALTV 44337	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
6		eqidd 2822	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)))
7		eqidd 2822	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))))
8	2, 3, 5, 6, 7	ringcvalALTV 44298	. . . 4 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩})
9	8	fveq2d 6674	. . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}))
10	1, 9	syl5eq 2868	. 2 ⊢ (𝜑 → 𝐻 = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}))
11	4	fvexi 6684	. . . 4 ⊢ 𝐵 ∈ V
12	11, 11	mpoex 7777	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) ∈ V
13		catstr 17227	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩} Struct ⟨1, ;15⟩
14		homid 16688	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
15		snsstp2 4750	. . . 4 ⊢ {⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}
16	13, 14, 15	strfv 16531	. . 3 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}))
17	12, 16	mp1i 13	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) = (Hom ‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}))
18	10, 17	eqtr4d 2859	1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3494  {ctp 4571  ⟨cop 4573   × cxp 5553   ∘ ccom 5559  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  1^st c1st 7687  2^nd c2nd 7688  1c1 10538  5c5 11696  ;cdc 12099  ndxcnx 16480  Basecbs 16483  Hom chom 16576  compcco 16577   RingHom crh 19464  RingCatALTVcringcALTV 44295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-fz 12894  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-hom 16589  df-cco 16590  df-ringcALTV 44297

This theorem is referenced by:  ringchomALTV  44339  ringccofvalALTV  44341