Theorem rngchomfvalALTV 49506

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 49505	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
6		eqidd 2764	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))
7		eqidd 2764	. . . . 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 49504	. . . 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 2810	. 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 8103	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)) ∈ V
13		catstr 18150	. . . 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 17598	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
15		snsstp2 4776	. . . 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 17396	. . 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 14	. 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 2801	1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  Vcvv 3454  {ctp 4592  ⟨cop 4838   × cxp 5653   ∘ ccom 5659  ‘cfv 6547  (class class class)co 7430   ∈ cmpo 7432  1^st c1st 8009  2^nd c2nd 8010  1c1 11214  5c5 12413  ;cdc 12827  ndxcnx 17386  Basecbs 17402  Hom chom 17454  compcco 17455   RngHom crnghm 20679  RngCatALTVcrngcALTV 49502

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-er 8731  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-5 12421  df-6 12422  df-7 12423  df-8 12424  df-9 12425  df-n0 12620  df-z 12707  df-dec 12828  df-uz 12979  df-fz 13653  df-struct 17340  df-slot 17375  df-ndx 17387  df-base 17403  df-hom 17467  df-cco 17468  df-rngcALTV 49503

This theorem is used by:  rngchomALTV  49507  rngccofvalALTV  49509  rngchomffvalALTV  49517