Theorem iinfconstbas 49425

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 49424	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝐴)
8	7	ne0d 4296	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ ∅)
9		iinconst 4959	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∩ ℎ ∈ 𝐴 𝑆 = 𝑆)
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → ∩ ℎ ∈ 𝐴 𝑆 = 𝑆)
11	10	eqcomd 2743	. . . 4 ⊢ (𝜑 → 𝑆 = ∩ ℎ ∈ 𝐴 𝑆)
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 = ∩ ℎ ∈ 𝐴 𝑆)
13	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐽 ∈ 𝐴)
14		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → ℎ = 𝐽)
15	14	oveqd 7385	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → (𝑥ℎ𝑦) = (𝑥𝐽𝑦))
16		snex 5385	. . . . . . . . . 10 ⊢ {(𝐼‘𝑥)} ∈ V
17		0ex 5254	. . . . . . . . . 10 ⊢ ∅ ∈ V
18	16, 17	ifex 4532	. . . . . . . . 9 ⊢ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ∈ V
19	1	ovmpt4g 7515	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ∈ V) → (𝑥𝐽𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
20	18, 19	mp3an3 1453	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐽𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
21	20	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → (𝑥𝐽𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
22	15, 21	eqtrd 2772	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → (𝑥ℎ𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
23		sseq1 3961	. . . . . . 7 ⊢ ({(𝐼‘𝑥)} = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) → ({(𝐼‘𝑥)} ⊆ (𝑥ℎ𝑦) ↔ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ⊆ (𝑥ℎ𝑦)))
24		sseq1 3961	. . . . . . 7 ⊢ (∅ = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) → (∅ ⊆ (𝑥ℎ𝑦) ↔ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ⊆ (𝑥ℎ𝑦)))
25		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ 𝐴)
26	6	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))
27	25, 26	eleqtrd 2839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))
28	27	elin1d 4158	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ (Subcat‘𝐶))
29	28	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → ℎ ∈ (Subcat‘𝐶))
30	27	elin2d 4159	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)})
31		vex 3446	. . . . . . . . . . . . . 14 ⊢ ℎ ∈ V
32		fneq1 6591	. . . . . . . . . . . . . 14 ⊢ (𝑗 = ℎ → (𝑗 Fn (𝑆 × 𝑆) ↔ ℎ Fn (𝑆 × 𝑆)))
33	31, 32	elab 3636	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)} ↔ ℎ Fn (𝑆 × 𝑆))
34	30, 33	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ Fn (𝑆 × 𝑆))
35	34	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → ℎ Fn (𝑆 × 𝑆))
36		simplrl 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → 𝑥 ∈ 𝑆)
37	29, 35, 36, 3	subcidcl 17780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → (𝐼‘𝑥) ∈ (𝑥ℎ𝑥))
38	37	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝐼‘𝑥) ∈ (𝑥ℎ𝑥))
39		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
40	39	oveq2d 7384	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝑥ℎ𝑥) = (𝑥ℎ𝑦))
41	38, 40	eleqtrd 2839	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝐼‘𝑥) ∈ (𝑥ℎ𝑦))
42	41	snssd 4767	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → {(𝐼‘𝑥)} ⊆ (𝑥ℎ𝑦))
43		0ss 4354	. . . . . . . 8 ⊢ ∅ ⊆ (𝑥ℎ𝑦)
44	43	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ ¬ 𝑥 = 𝑦) → ∅ ⊆ (𝑥ℎ𝑦))
45	23, 24, 42, 44	ifbothda 4520	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ⊆ (𝑥ℎ𝑦))
46	13, 22, 45	iinglb 49181	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
47	46	eqcomd 2743	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) = ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦))
48	11, 12, 47	mpoeq123dva 7442	. . 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 2784	. 2 ⊢ (𝜑 → 𝐽 = (𝑥 ∈ ∩ ℎ ∈ 𝐴 𝑆, 𝑦 ∈ ∩ ℎ ∈ 𝐴 𝑆 ↦ ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦)))
50		eqid 2737	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
51	28, 50	subcssc 17776	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ⊆cat (Homf ‘𝐶))
52		eqidd 2738	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)) = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)))
53		dmdm 49412	. . . 4 ⊢ (ℎ Fn (𝑆 × 𝑆) → 𝑆 = dom dom ℎ)
54	34, 53	syl 17	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → 𝑆 = dom dom ℎ)
55		nfv 1916	. . 3 ⊢ Ⅎℎ𝜑
56	8, 51, 52, 54, 55	iinfssclem1 49413	. 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 2775	1 ⊢ (𝜑 → 𝐽 = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  ifcif 4481  {csn 4582  ∩ ciin 4949   ↦ cmpt 5181   × cxp 5630  dom cdm 5632   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Basecbs 17148  Catccat 17599  Idccid 17600  Hom_f chomf 17601  Subcatcsubc 17745

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-pm 8778  df-ixp 8848  df-cat 17603  df-cid 17604  df-homf 17605  df-ssc 17746  df-subc 17748

This theorem is referenced by: (None)