Theorem infsubc 49838

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

Assertion

Ref	Expression
infsubc	⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) ∈ (Subcat‘𝐶))

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

Proof of Theorem infsubc

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prnzg 4744	. . 3 ⊢ (𝐴 ∈ (Subcat‘𝐶) → {𝐴, 𝐵} ≠ ∅)
2	1	adantr 485	. 2 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → {𝐴, 𝐵} ≠ ∅)
3		simpll 778	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝐴 ∈ (Subcat‘𝐶))
4		eleq1 2851	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (Subcat‘𝐶) ↔ 𝐴 ∈ (Subcat‘𝐶)))
5	3, 4	syl5ibrcom 250	. . 3 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → (𝑦 = 𝐴 → 𝑦 ∈ (Subcat‘𝐶)))
6		simplr 780	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝐵 ∈ (Subcat‘𝐶))
7		eleq1 2851	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ (Subcat‘𝐶) ↔ 𝐵 ∈ (Subcat‘𝐶)))
8	6, 7	syl5ibrcom 250	. . 3 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → (𝑦 = 𝐵 → 𝑦 ∈ (Subcat‘𝐶)))
9		elpri 4613	. . . 4 ⊢ (𝑦 ∈ {𝐴, 𝐵} → (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
10	9	adantl 486	. . 3 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
11	5, 8, 10	mpjaod 873	. 2 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝑦 ∈ (Subcat‘𝐶))
12		iinfprg 49837	. 2 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) = (𝑥 ∈ ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 ↦ ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥)))
13	2, 11, 12	iinfsubc 49836	1 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   ∩ cin 3904  ∅c0 4286  {cpr 4591   ↦ cmpt 5192  dom cdm 5661  ‘cfv 6536  Subcatcsubc 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-pm 8823  df-ixp 8892  df-ssc 17862  df-subc 17864

This theorem is referenced by:  infsubc2  49839