Theorem iinfconstbas 50319

Description: The discrete category is the indexed intersection of all subcategories with the same base. (Contributed by Zhi Wang, 1-Nov-2025.)

Hypotheses

Ref	Expression
discsubc.j	⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
discsubc.b	⊢ 𝐵 = (Base‘𝐶)
discsubc.i	⊢ 𝐼 = (Id‘𝐶)
discsubc.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
discsubc.c	⊢ (𝜑 → 𝐶 ∈ Cat)
iinfconstbas.a	⊢ (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))

Assertion

Ref	Expression
iinfconstbas	⊢ (𝜑 → 𝐽 = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)))

Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐼,𝑦   ℎ,𝐽,𝑗   𝑆,ℎ,𝑗   𝐴,ℎ,𝑥,𝑦,𝑧   ℎ,𝐼   𝑧,𝑆   𝜑,ℎ,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧, 𝑗)   𝐴(𝑗)   𝐵(𝑥, 𝑦, 𝑧, ℎ, 𝑗)   𝐶(𝑥, 𝑦, 𝑧, ℎ, 𝑗)   𝐼(𝑧, 𝑗)   𝐽(𝑥, 𝑦, 𝑧)

Proof of Theorem iinfconstbas

Step	Hyp	Ref	Expression
1		discsubc.j	. . 3 ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
2		discsubc.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
3		discsubc.i	. . . . . . . 8 ⊢ 𝐼 = (Id‘𝐶)
4		discsubc.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
5		discsubc.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		iinfconstbas.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))
7	1, 2, 3, 4, 5, 6	iinfconstbaslem 50318	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝐴)
8	7	ne0d 4290	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ ∅)
9		iinconst 4964	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∩ ℎ ∈ 𝐴 𝑆 = 𝑆)
10	8, 9	syl 18	. . . . 5 ⊢ (𝜑 → ∩ ℎ ∈ 𝐴 𝑆 = 𝑆)
11	10	eqcomd 2769	. . . 4 ⊢ (𝜑 → 𝑆 = ∩ ℎ ∈ 𝐴 𝑆)
12	11	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 = ∩ ℎ ∈ 𝐴 𝑆)
13	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐽 ∈ 𝐴)
14		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → ℎ = 𝐽)
15	14	oveqd 7444	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → (𝑥ℎ𝑦) = (𝑥𝐽𝑦))
16		snex 5400	. . . . . . . . . 10 ⊢ {(𝐼‘𝑥)} ∈ V
17		0ex 5263	. . . . . . . . . 10 ⊢ ∅ ∈ V
18	16, 17	ifex 4535	. . . . . . . . 9 ⊢ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ∈ V
19	1	ovmpt4g 7574	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ∈ V) → (𝑥𝐽𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
20	18, 19	mp3an3 1479	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐽𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
21	20	ad2antlr 740	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → (𝑥𝐽𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
22	15, 21	eqtrd 2798	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → (𝑥ℎ𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
23		sseq1 3937	. . . . . . 7 ⊢ ({(𝐼‘𝑥)} = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) → ({(𝐼‘𝑥)} ⊆ (𝑥ℎ𝑦) ↔ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ⊆ (𝑥ℎ𝑦)))
24		sseq1 3937	. . . . . . 7 ⊢ (∅ = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) → (∅ ⊆ (𝑥ℎ𝑦) ↔ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ⊆ (𝑥ℎ𝑦)))
25		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ 𝐴)
26	6	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))
27	25, 26	eleqtrd 2865	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))
28	27	elin1d 4103	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ (Subcat‘𝐶))
29	28	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → ℎ ∈ (Subcat‘𝐶))
30	27	elin2d 4104	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)})
31		vex 3457	. . . . . . . . . . . . . 14 ⊢ ℎ ∈ V
32		fneq1 6638	. . . . . . . . . . . . . 14 ⊢ (𝑗 = ℎ → (𝑗 Fn (𝑆 × 𝑆) ↔ ℎ Fn (𝑆 × 𝑆)))
33	31, 32	elab 3635	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)} ↔ ℎ Fn (𝑆 × 𝑆))
34	30, 33	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ Fn (𝑆 × 𝑆))
35	34	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → ℎ Fn (𝑆 × 𝑆))
36		simplrl 789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → 𝑥 ∈ 𝑆)
37	29, 35, 36, 3	subcidcl 18031	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → (𝐼‘𝑥) ∈ (𝑥ℎ𝑥))
38	37	adantr 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝐼‘𝑥) ∈ (𝑥ℎ𝑥))
39		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
40	39	oveq2d 7443	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝑥ℎ𝑥) = (𝑥ℎ𝑦))
41	38, 40	eleqtrd 2865	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝐼‘𝑥) ∈ (𝑥ℎ𝑦))
42	41	snssd 4746	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → {(𝐼‘𝑥)} ⊆ (𝑥ℎ𝑦))
43		0ss 4352	. . . . . . . 8 ⊢ ∅ ⊆ (𝑥ℎ𝑦)
44	43	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ ¬ 𝑥 = 𝑦) → ∅ ⊆ (𝑥ℎ𝑦))
45	23, 24, 42, 44	ifbothda 4523	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ⊆ (𝑥ℎ𝑦))
46	13, 22, 45	iinglb 50072	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
47	46	eqcomd 2769	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) = ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦))
48	11, 12, 47	mpoeq123dva 7501	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) = (𝑥 ∈ ∩ ℎ ∈ 𝐴 𝑆, 𝑦 ∈ ∩ ℎ ∈ 𝐴 𝑆 ↦ ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦)))
49	1, 48	eqtrid 2810	. 2 ⊢ (𝜑 → 𝐽 = (𝑥 ∈ ∩ ℎ ∈ 𝐴 𝑆, 𝑦 ∈ ∩ ℎ ∈ 𝐴 𝑆 ↦ ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦)))
50		eqid 2763	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
51	28, 50	subcssc 18027	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ⊆cat (Homf ‘𝐶))
52		eqidd 2764	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)) = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)))
53		dmdm 50306	. . . 4 ⊢ (ℎ Fn (𝑆 × 𝑆) → 𝑆 = dom dom ℎ)
54	34, 53	syl 18	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → 𝑆 = dom dom ℎ)
55		nfv 1948	. . 3 ⊢ Ⅎℎ𝜑
56	8, 51, 52, 54, 55	iinfssclem1 50307	. 2 ⊢ (𝜑 → (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)) = (𝑥 ∈ ∩ ℎ ∈ 𝐴 𝑆, 𝑦 ∈ ∩ ℎ ∈ 𝐴 𝑆 ↦ ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦)))
57	49, 56	eqtr4d 2801	1 ⊢ (𝜑 → 𝐽 = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {cab 2741   ≠ wne 2958  Vcvv 3453   ⊆ wss 3899   ∩ cin 4091  ∅c0 4280  ifcif 4484  {csn 4587  ∩ ciin 4954   ↦ cmpt 5188   × cxp 5652  dom cdm 5654   Fn wfn 6541  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429  Basecbs 17399  Catccat 17850  Idccid 17851  Hom_f chomf 17852  Subcatcsubc 17996

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 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-iin 4956  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-pm 8861  df-ixp 8937  df-cat 17854  df-cid 17855  df-homf 17856  df-ssc 17997  df-subc 17999

This theorem is used by: (None)