Theorem ringcbasALTV 48776

Description: Set of objects of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
ringcbasALTV	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))

Proof of Theorem ringcbasALTV

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

Step	Hyp	Ref	Expression
1		ringcbasALTV.c	. . 3 ⊢ 𝐶 = (RingCatALTV‘𝑈)
2		ringcbasALTV.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		eqidd 2737	. . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) = (𝑈 ∩ Ring))
4		eqidd 2737	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)) = (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦)))
5		eqidd 2737	. . 3 ⊢ (𝜑 → (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))) = (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))))
6	1, 2, 3, 4, 5	ringcvalALTV 48765	. 2 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Ring)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩})
7		catstr 17927	. 2 ⊢ {⟨(Base‘ndx), (𝑈 ∩ Ring)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩} Struct ⟨1, ;15⟩
8		baseid 17182	. 2 ⊢ Base = Slot (Base‘ndx)
9		snsstp1 4759	. 2 ⊢ {⟨(Base‘ndx), (𝑈 ∩ Ring)⟩} ⊆ {⟨(Base‘ndx), (𝑈 ∩ Ring)⟩, ⟨(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Ring), 𝑦 ∈ (𝑈 ∩ Ring) ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)), 𝑧 ∈ (𝑈 ∩ Ring) ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))⟩}
10		inex1g 5260	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
11	2, 10	syl 17	. 2 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V)
12		ringcbasALTV.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
13	6, 7, 8, 9, 11, 12	strfv3 17174	1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888  {ctp 4571  ⟨cop 4573   × cxp 5629   ∘ ccom 5635  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941  1c1 11039  5c5 12239  ;cdc 12644  ndxcnx 17163  Basecbs 17179  Hom chom 17231  compcco 17232  Ringcrg 20214   RingHom crh 20449  RingCatALTVcringcALTV 48763

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-ringcALTV 48764

This theorem is referenced by:  ringchomfvalALTV  48777  ringccofvalALTV  48780  ringccatidALTV  48782  ringcbasbasALTV  48788  funcringcsetclem7ALTV  48795  srhmsubcALTVlem1  48799  srhmsubcALTV  48801