Theorem infsubc 49045

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 4730	. . 3 ⊢ (𝐴 ∈ (Subcat‘𝐶) → {𝐴, 𝐵} ≠ ∅)
2	1	adantr 480	. 2 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → {𝐴, 𝐵} ≠ ∅)
3		simpll 766	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝐴 ∈ (Subcat‘𝐶))
4		eleq1 2816	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (Subcat‘𝐶) ↔ 𝐴 ∈ (Subcat‘𝐶)))
5	3, 4	syl5ibrcom 247	. . 3 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → (𝑦 = 𝐴 → 𝑦 ∈ (Subcat‘𝐶)))
6		simplr 768	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝐵 ∈ (Subcat‘𝐶))
7		eleq1 2816	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ (Subcat‘𝐶) ↔ 𝐵 ∈ (Subcat‘𝐶)))
8	6, 7	syl5ibrcom 247	. . 3 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → (𝑦 = 𝐵 → 𝑦 ∈ (Subcat‘𝐶)))
9		elpri 4601	. . . 4 ⊢ (𝑦 ∈ {𝐴, 𝐵} → (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
10	9	adantl 481	. . 3 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
11	5, 8, 10	mpjaod 860	. 2 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝑦 ∈ (Subcat‘𝐶))
12		iinfprg 49044	. 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 49043	1 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∩ cin 3902  ∅c0 4284  {cpr 4579   ↦ cmpt 5173  dom cdm 5619  ‘cfv 6482  Subcatcsubc 17716

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-pm 8756  df-ixp 8825  df-ssc 17717  df-subc 17719

This theorem is referenced by:  infsubc2  49046