Theorem infsubc2 49538

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 4723	. . . . 5 ⊢ (𝐴 ∈ (Subcat‘𝐶) → {𝐴, 𝐵} ≠ ∅)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → {𝐴, 𝐵} ≠ ∅)
3		simpll 767	. . . . . . 7 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐴 ∈ (Subcat‘𝐶))
4		eqid 2737	. . . . . . 7 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5	3, 4	subcssc 17809	. . . . . 6 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐴 ⊆cat (Homf ‘𝐶))
6		breq1 5089	. . . . . 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 769	. . . . . . 7 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐵 ∈ (Subcat‘𝐶))
9	8, 4	subcssc 17809	. . . . . 6 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝐵 ⊆cat (Homf ‘𝐶))
10		breq1 5089	. . . . . 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 4592	. . . . . 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 861	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → 𝑤 ⊆cat (Homf ‘𝐶))
15		iinfprg 49536	. . . 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 2738	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑤 ∈ {𝐴, 𝐵}) → dom dom 𝑤 = dom dom 𝑤)
17		nfv 1916	. . . 4 ⊢ Ⅎ𝑤(𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶))
18	2, 14, 15, 16, 17	iinfssclem1 49531	. . 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 5860	. . . . . 6 ⊢ (𝑤 = 𝐴 → dom 𝑤 = dom 𝐴)
20	19	dmeqd 5862	. . . . 5 ⊢ (𝑤 = 𝐴 → dom dom 𝑤 = dom dom 𝐴)
21		dmeq 5860	. . . . . 6 ⊢ (𝑤 = 𝐵 → dom 𝑤 = dom 𝐵)
22	21	dmeqd 5862	. . . . 5 ⊢ (𝑤 = 𝐵 → dom dom 𝑤 = dom dom 𝐵)
23	20, 22	iinxprg 5032	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → ∩ 𝑤 ∈ {𝐴, 𝐵}dom dom 𝑤 = (dom dom 𝐴 ∩ dom dom 𝐵))
24		oveq 7375	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑥𝑤𝑦) = (𝑥𝐴𝑦))
25		oveq 7375	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝑥𝑤𝑦) = (𝑥𝐵𝑦))
26	24, 25	iinxprg 5032	. . . 4 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → ∩ 𝑤 ∈ {𝐴, 𝐵} (𝑥𝑤𝑦) = ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦)))
27	23, 23, 26	mpoeq123dv 7444	. . 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 2772	. 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 49537	. 2 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑧 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑧) ∩ (𝐵‘𝑧))) ∈ (Subcat‘𝐶))
30	28, 29	eqeltrrd 2838	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 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∩ cin 3889  ∅c0 4274  {cpr 4570  ∩ ciin 4935   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5632  ‘cfv 6500  (class class class)co 7369   ∈ cmpo 7371  Hom_f chomf 17634   ⊆_cat cssc 17776  Subcatcsubc 17778

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7691

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-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  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-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7944  df-2nd 7945  df-pm 8778  df-ixp 8848  df-ssc 17779  df-subc 17781

This theorem is referenced by:  infsubc2d  49539