Theorem infsubc 50167

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 4739	. . 3 ⊢ (𝐴 ∈ (Subcat‘𝐶) → {𝐴, 𝐵} ≠ ∅)
2	1	adantr 486	. 2 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → {𝐴, 𝐵} ≠ ∅)
3		simpll 779	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝐴 ∈ (Subcat‘𝐶))
4		eleq1 2849	. . . 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 781	. . . 4 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝐵 ∈ (Subcat‘𝐶))
7		eleq1 2849	. . . 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 4608	. . . 4 ⊢ (𝑦 ∈ {𝐴, 𝐵} → (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
10	9	adantl 487	. . 3 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
11	5, 8, 10	mpjaod 874	. 2 ⊢ (((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝑦 ∈ (Subcat‘𝐶))
12		iinfprg 50166	. 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 50165	1 ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) ∈ (Subcat‘𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   ∩ cin 3898  ∅c0 4279  {cpr 4586   ↦ cmpt 5186  dom cdm 5651  ‘cfv 6538  Subcatcsubc 17984

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-pm 8850  df-ixp 8926  df-ssc 17985  df-subc 17987

This theorem is used by:  infsubc2  50168