Theorem infsubc2d 49897

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

Hypotheses

Ref	Expression
infsubc2d.1	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
infsubc2d.2	⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
infsubc2d.3	⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))
infsubc2d.4	⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))

Assertion

Ref	Expression
infsubc2d	⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))

Distinct variable groups:   𝑥,𝐶,𝑦   𝑥,𝐻,𝑦   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem infsubc2d

Step	Hyp	Ref	Expression
1		infsubc2d.1	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
2	1	fndmd 6644	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
3	2	dmeqd 5897	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
4		dmxpid 5922	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
5	3, 4	eqtrdi 2816	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
6		infsubc2d.2	. . . . . . 7 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
7	6	fndmd 6644	. . . . . 6 ⊢ (𝜑 → dom 𝐽 = (𝑇 × 𝑇))
8	7	dmeqd 5897	. . . . 5 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑇 × 𝑇))
9		dmxpid 5922	. . . . 5 ⊢ dom (𝑇 × 𝑇) = 𝑇
10	8, 9	eqtrdi 2816	. . . 4 ⊢ (𝜑 → dom dom 𝐽 = 𝑇)
11	5, 10	ineq12d 4174	. . 3 ⊢ (𝜑 → (dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆 ∩ 𝑇))
12		mpoeq12 7492	. . 3 ⊢ (((dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆 ∩ 𝑇) ∧ (dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆 ∩ 𝑇)) → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) = (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))))
13	11, 11, 12	syl2anc 596	. 2 ⊢ (𝜑 → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) = (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))))
14		infsubc2d.3	. . 3 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))
15		infsubc2d.4	. . 3 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
16		infsubc2 49896	. . 3 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
17	14, 15, 16	syl2anc 596	. 2 ⊢ (𝜑 → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
18	13, 17	eqeltrrd 2866	1 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ∩ cin 3905   × cxp 5661  dom cdm 5663   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  Subcatcsubc 17888

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-pm 8833  df-ixp 8902  df-ssc 17889  df-subc 17891

This theorem is used by: (None)