Theorem dfrngc2 42297

Description: Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)

Hypotheses

Ref	Expression
dfrngc2.c	⊢ 𝐶 = (RngCat‘𝑈)
dfrngc2.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
dfrngc2.b	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
dfrngc2.h	⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
dfrngc2.o	⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈)))

Assertion

Ref	Expression
dfrngc2	⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Proof of Theorem dfrngc2

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

Step	Hyp	Ref	Expression
1		dfrngc2.c	. . 3 ⊢ 𝐶 = (RngCat‘𝑈)
2		dfrngc2.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		dfrngc2.b	. . 3 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
4		dfrngc2.h	. . 3 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
5	1, 2, 3, 4	rngcval 42287	. 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
6		eqid 2651	. . . 4 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻)
7		fvexd 6241	. . . 4 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V)
8		inex1g 4834	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
9	2, 8	syl 17	. . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
10	3, 9	eqeltrd 2730	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
11	3, 4	rnghmresfn 42288	. . . 4 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
12	6, 7, 10, 11	rescval2 16535	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
13		eqid 2651	. . . . 5 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
14		eqidd 2652	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
15		dfrngc2.o	. . . . . 6 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈)))
16		eqid 2651	. . . . . . 7 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
17	13, 2, 16	estrccofval 16816	. . . . . 6 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
18	15, 17	eqtrd 2685	. . . . 5 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
19	13, 2, 14, 18	estrcval 16811	. . . 4 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
20	2, 2	jca 553	. . . . 5 ⊢ (𝜑 → (𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉))
21		eqid 2651	. . . . . 6 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
22	21	mpt2exg 7290	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
23	20, 22	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
24		fvexd 6241	. . . . 5 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
25	15, 24	eqeltrd 2730	. . . 4 ⊢ (𝜑 → · ∈ V)
26		rnghmfn 42215	. . . . . . 7 ⊢ RngHomo Fn (Rng × Rng)
27		fnfun 6026	. . . . . . 7 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
28	26, 27	mp1i 13	. . . . . 6 ⊢ (𝜑 → Fun RngHomo )
29		sqxpexg 7005	. . . . . . 7 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
30	10, 29	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
31		resfunexg 6520	. . . . . 6 ⊢ ((Fun RngHomo ∧ (𝐵 × 𝐵) ∈ V) → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
32	28, 30, 31	syl2anc 694	. . . . 5 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
33	4, 32	eqeltrd 2730	. . . 4 ⊢ (𝜑 → 𝐻 ∈ V)
34		inss1 3866	. . . . . 6 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
35	34	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈)
36	3	sseq1d 3665	. . . . 5 ⊢ (𝜑 → (𝐵 ⊆ 𝑈 ↔ (𝑈 ∩ Rng) ⊆ 𝑈))
37	35, 36	mpbird 247	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
38	19, 2, 23, 25, 10, 33, 37	estrres 16826	. . 3 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
39	12, 38	eqtrd 2685	. 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
40	5, 39	eqtrd 2685	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  {ctp 4214  ⟨cop 4216   × cxp 5141   ↾ cres 5145   ∘ ccom 5147  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  1^st c1st 7208  2^nd c2nd 7209   ↑_𝑚 cmap 7899  ndxcnx 15901   sSet csts 15902  Basecbs 15904   ↾_s cress 15905  Hom chom 15999  compcco 16000   ↾_cat cresc 16515  ExtStrCatcestrc 16809  Rngcrng 42199   RngHomo crngh 42210  RngCatcrngc 42282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-hom 16013  df-cco 16014  df-resc 16518  df-estrc 16810  df-rnghomo 42212  df-rngc 42284

This theorem is referenced by:  rngcresringcat  42355