Theorem infsubc2 49723

Description: The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.)

Assertion

Ref	Expression
infsubc2	⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom dom 𝐴 ∩ dom dom 𝐵), 𝑦 ∈ (dom dom 𝐴 ∩ dom dom 𝐵) ↦ ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦))) ∈ (Subcat‘𝐶))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem infsubc2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prnzg 4749	. . . . 5 ⊢ (𝐴 ∈ (Subcat‘𝐶) → {𝐴, 𝐵} ≠ ∅)
2	1	adantr 485	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → {𝐴, 𝐵} ≠ ∅)
3		simpll 778	. . . . . . 7 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐴 ∈ (Subcat‘𝐶))
4		eqid 2769	. . . . . . 7 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5	3, 4	subcssc 17896	. . . . . 6 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐴 ⊆cat (Homf ‘𝐶))
6		breq1 5116	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆cat (Homf ‘𝐶) ↔ 𝐴 ⊆cat (Homf ‘𝐶)))
7	5, 6	syl5ibrcom 250	. . . . 5 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → (𝑤 = 𝐴 → 𝑤 ⊆cat (Homf ‘𝐶)))
8		simplr 780	. . . . . . 7 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐵 ∈ (Subcat‘𝐶))
9	8, 4	subcssc 17896	. . . . . 6 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐵 ⊆cat (Homf ‘𝐶))
10		breq1 5116	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ⊆cat (Homf ‘𝐶) ↔ 𝐵 ⊆cat (Homf ‘𝐶)))
11	9, 10	syl5ibrcom 250	. . . . 5 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → (𝑤 = 𝐵 → 𝑤 ⊆cat (Homf ‘𝐶)))
12		elpri 4618	. . . . . 6 ⊢ (𝑤 ∈ {𝐴, 𝐵} → (𝑤 = 𝐴 ∨ 𝑤 = 𝐵))
13	12	adantl 486	. . . . 5 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → (𝑤 = 𝐴 ∨ 𝑤 = 𝐵))
14	7, 11, 13	mpjaod 873	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝑤 ⊆cat (Homf ‘𝐶))
15		iinfprg 49721	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑧 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑧) ∩ (𝐵‘𝑧))) = (𝑧 ∈ ∩ 𝑤 ∈ {𝐴, 𝐵}dom 𝑤 ↦ ∩ 𝑤 ∈ {𝐴, 𝐵} (𝑤‘𝑧)))
16		eqidd 2770	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → dom dom 𝑤 = dom dom 𝑤)
17		nfv 1941	. . . 4 ⊢ Ⅎ𝑤(𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶))
18	2, 14, 15, 16, 17	iinfssclem1 49716	. . 3 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑧 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑧) ∩ (𝐵‘𝑧))) = (𝑥 ∈ ∩ 𝑤 ∈ {𝐴, 𝐵}dom dom 𝑤, 𝑦 ∈ ∩ 𝑤 ∈ {𝐴, 𝐵}dom dom 𝑤 ↦ ∩ 𝑤 ∈ {𝐴, 𝐵} (𝑥𝑤𝑦)))
19		dmeq 5894	. . . . . 6 ⊢ (𝑤 = 𝐴 → dom 𝑤 = dom 𝐴)
20	19	dmeqd 5896	. . . . 5 ⊢ (𝑤 = 𝐴 → dom dom 𝑤 = dom dom 𝐴)
21		dmeq 5894	. . . . . 6 ⊢ (𝑤 = 𝐵 → dom 𝑤 = dom 𝐵)
22	21	dmeqd 5896	. . . . 5 ⊢ (𝑤 = 𝐵 → dom dom 𝑤 = dom dom 𝐵)
23	20, 22	iinxprg 5059	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → ∩ 𝑤 ∈ {𝐴, 𝐵}dom dom 𝑤 = (dom dom 𝐴 ∩ dom dom 𝐵))
24		oveq 7417	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑥𝑤𝑦) = (𝑥𝐴𝑦))
25		oveq 7417	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝑥𝑤𝑦) = (𝑥𝐵𝑦))
26	24, 25	iinxprg 5059	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → ∩ 𝑤 ∈ {𝐴, 𝐵} (𝑥𝑤𝑦) = ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦)))
27	23, 23, 26	mpoeq123dv 7486	. . 3 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ ∩ 𝑤 ∈ {𝐴, 𝐵}dom dom 𝑤, 𝑦 ∈ ∩ 𝑤 ∈ {𝐴, 𝐵}dom dom 𝑤 ↦ ∩ 𝑤 ∈ {𝐴, 𝐵} (𝑥𝑤𝑦)) = (𝑥 ∈ (dom dom 𝐴 ∩ dom dom 𝐵), 𝑦 ∈ (dom dom 𝐴 ∩ dom dom 𝐵) ↦ ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦))))
28	18, 27	eqtrd 2804	. 2 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑧 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑧) ∩ (𝐵‘𝑧))) = (𝑥 ∈ (dom dom 𝐴 ∩ dom dom 𝐵), 𝑦 ∈ (dom dom 𝐴 ∩ dom dom 𝐵) ↦ ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦))))
29		infsubc 49722	. 2 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑧 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑧) ∩ (𝐵‘𝑧))) ∈ (Subcat‘𝐶))
30	28, 29	eqeltrrd 2870	1 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom dom 𝐴 ∩ dom dom 𝐵), 𝑦 ∈ (dom dom 𝐴 ∩ dom dom 𝐵) ↦ ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦))) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∩ cin 3912  ∅c0 4294  {cpr 4596  ∩ ciin 4961   class class class wbr 5113   ↦ cmpt 5196  dom cdm 5662  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  Hom_f chomf 17721   ⊆_cat cssc 17863  Subcatcsubc 17865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-pm 8826  df-ixp 8895  df-ssc 17866  df-subc 17868

This theorem is referenced by:  infsubc2d  49724