Theorem iinfconstbas 49687

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 49686	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝐴)
8	7	ne0d 4294	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ ∅)
9		iinconst 4960	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∩ ℎ ∈ 𝐴 𝑆 = 𝑆)
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → ∩ ℎ ∈ 𝐴 𝑆 = 𝑆)
11	10	eqcomd 2768	. . . 4 ⊢ (𝜑 → 𝑆 = ∩ ℎ ∈ 𝐴 𝑆)
12	11	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 = ∩ ℎ ∈ 𝐴 𝑆)
13	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐽 ∈ 𝐴)
14		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → ℎ = 𝐽)
15	14	oveqd 7413	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → (𝑥ℎ𝑦) = (𝑥𝐽𝑦))
16		snex 5396	. . . . . . . . . 10 ⊢ {(𝐼‘𝑥)} ∈ V
17		0ex 5257	. . . . . . . . . 10 ⊢ ∅ ∈ V
18	16, 17	ifex 4531	. . . . . . . . 9 ⊢ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ∈ V
19	1	ovmpt4g 7543	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ∈ V) → (𝑥𝐽𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
20	18, 19	mp3an3 1471	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐽𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
21	20	ad2antlr 737	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ = 𝐽) → (𝑥𝐽𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
22	15, 21	eqtrd 2797	. . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ 𝐴)
26	6	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))
27	25, 26	eleqtrd 2864	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))
28	27	elin1d 4156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ (Subcat‘𝐶))
29	28	adantlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → ℎ ∈ (Subcat‘𝐶))
30	27	elin2d 4157	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ∈ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)})
31		vex 3458	. . . . . . . . . . . . . 14 ⊢ ℎ ∈ V
32		fneq1 6612	. . . . . . . . . . . . . 14 ⊢ (𝑗 = ℎ → (𝑗 Fn (𝑆 × 𝑆) ↔ ℎ Fn (𝑆 × 𝑆)))
33	31, 32	elab 3638	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)} ↔ ℎ Fn (𝑆 × 𝑆))
34	30, 33	sylib 220	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ Fn (𝑆 × 𝑆))
35	34	adantlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → ℎ Fn (𝑆 × 𝑆))
36		simplrl 786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → 𝑥 ∈ 𝑆)
37	29, 35, 36, 3	subcidcl 17877	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → (𝐼‘𝑥) ∈ (𝑥ℎ𝑥))
38	37	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝐼‘𝑥) ∈ (𝑥ℎ𝑥))
39		simpr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
40	39	oveq2d 7412	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝑥ℎ𝑥) = (𝑥ℎ𝑦))
41	38, 40	eleqtrd 2864	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝐼‘𝑥) ∈ (𝑥ℎ𝑦))
42	41	snssd 4745	. . . . . . 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 4519	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ ℎ ∈ 𝐴) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ⊆ (𝑥ℎ𝑦))
46	13, 22, 45	iinglb 49443	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦) = if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))
47	46	eqcomd 2768	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) = ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦))
48	11, 12, 47	mpoeq123dva 7470	. . 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 2809	. 2 ⊢ (𝜑 → 𝐽 = (𝑥 ∈ ∩ ℎ ∈ 𝐴 𝑆, 𝑦 ∈ ∩ ℎ ∈ 𝐴 𝑆 ↦ ∩ ℎ ∈ 𝐴 (𝑥ℎ𝑦)))
50		eqid 2762	. . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
51	28, 50	subcssc 17873	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → ℎ ⊆cat (Homf ‘𝐶))
52		eqidd 2763	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)) = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)))
53		dmdm 49674	. . . 4 ⊢ (ℎ Fn (𝑆 × 𝑆) → 𝑆 = dom dom ℎ)
54	34, 53	syl 17	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐴) → 𝑆 = dom dom ℎ)
55		nfv 1934	. . 3 ⊢ Ⅎℎ𝜑
56	8, 51, 52, 54, 55	iinfssclem1 49675	. 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 2800	1 ⊢ (𝜑 → 𝐽 = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {cab 2740   ≠ wne 2957  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ifcif 4480  {csn 4582  ∩ ciin 4950   ↦ cmpt 5181   × cxp 5645  dom cdm 5647   Fn wfn 6516  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  Basecbs 17245  Catccat 17696  Idccid 17697  Hom_f chomf 17698  Subcatcsubc 17842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-pm 8811  df-ixp 8880  df-cat 17700  df-cid 17701  df-homf 17702  df-ssc 17843  df-subc 17845

This theorem is referenced by: (None)