Theorem infsubc2 49038

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 4744	. . . . 5 ⊢ (𝐴 ∈ (Subcat‘𝐶) → {𝐴, 𝐵} ≠ ∅)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → {𝐴, 𝐵} ≠ ∅)
3		simpll 766	. . . . . . 7 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐴 ∈ (Subcat‘𝐶))
4		eqid 2730	. . . . . . 7 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5	3, 4	subcssc 17808	. . . . . 6 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐴 ⊆cat (Homf ‘𝐶))
6		breq1 5112	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆cat (Homf ‘𝐶) ↔ 𝐴 ⊆cat (Homf ‘𝐶)))
7	5, 6	syl5ibrcom 247	. . . . 5 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → (𝑤 = 𝐴 → 𝑤 ⊆cat (Homf ‘𝐶)))
8		simplr 768	. . . . . . 7 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐵 ∈ (Subcat‘𝐶))
9	8, 4	subcssc 17808	. . . . . 6 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐵 ⊆cat (Homf ‘𝐶))
10		breq1 5112	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ⊆cat (Homf ‘𝐶) ↔ 𝐵 ⊆cat (Homf ‘𝐶)))
11	9, 10	syl5ibrcom 247	. . . . 5 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → (𝑤 = 𝐵 → 𝑤 ⊆cat (Homf ‘𝐶)))
12		elpri 4615	. . . . . 6 ⊢ (𝑤 ∈ {𝐴, 𝐵} → (𝑤 = 𝐴 ∨ 𝑤 = 𝐵))
13	12	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → (𝑤 = 𝐴 ∨ 𝑤 = 𝐵))
14	7, 11, 13	mpjaod 860	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝑤 ⊆cat (Homf ‘𝐶))
15		iinfprg 49036	. . . 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 2731	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → dom dom 𝑤 = dom dom 𝑤)
17		nfv 1914	. . . 4 ⊢ Ⅎ𝑤(𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶))
18	2, 14, 15, 16, 17	iinfssclem1 49031	. . 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 5869	. . . . . 6 ⊢ (𝑤 = 𝐴 → dom 𝑤 = dom 𝐴)
20	19	dmeqd 5871	. . . . 5 ⊢ (𝑤 = 𝐴 → dom dom 𝑤 = dom dom 𝐴)
21		dmeq 5869	. . . . . 6 ⊢ (𝑤 = 𝐵 → dom 𝑤 = dom 𝐵)
22	21	dmeqd 5871	. . . . 5 ⊢ (𝑤 = 𝐵 → dom dom 𝑤 = dom dom 𝐵)
23	20, 22	iinxprg 5055	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → ∩ 𝑤 ∈ {𝐴, 𝐵}dom dom 𝑤 = (dom dom 𝐴 ∩ dom dom 𝐵))
24		oveq 7395	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑥𝑤𝑦) = (𝑥𝐴𝑦))
25		oveq 7395	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝑥𝑤𝑦) = (𝑥𝐵𝑦))
26	24, 25	iinxprg 5055	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → ∩ 𝑤 ∈ {𝐴, 𝐵} (𝑥𝑤𝑦) = ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦)))
27	23, 23, 26	mpoeq123dv 7466	. . 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 2765	. 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 49037	. 2 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑧 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑧) ∩ (𝐵‘𝑧))) ∈ (Subcat‘𝐶))
30	28, 29	eqeltrrd 2830	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 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   ∩ cin 3915  ∅c0 4298  {cpr 4593  ∩ ciin 4958   class class class wbr 5109   ↦ cmpt 5190  dom cdm 5640  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  Hom_f chomf 17633   ⊆_cat cssc 17775  Subcatcsubc 17777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-iin 4960  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-pm 8804  df-ixp 8873  df-ssc 17778  df-subc 17780

This theorem is referenced by:  infsubc2d  49039