Theorem infsubc2d 49724

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 6641	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
3	2	dmeqd 5896	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
4		dmxpid 5921	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
5	3, 4	eqtrdi 2820	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
6		infsubc2d.2	. . . . . . 7 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
7	6	fndmd 6641	. . . . . 6 ⊢ (𝜑 → dom 𝐽 = (𝑇 × 𝑇))
8	7	dmeqd 5896	. . . . 5 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑇 × 𝑇))
9		dmxpid 5921	. . . . 5 ⊢ dom (𝑇 × 𝑇) = 𝑇
10	8, 9	eqtrdi 2820	. . . 4 ⊢ (𝜑 → dom dom 𝐽 = 𝑇)
11	5, 10	ineq12d 4182	. . 3 ⊢ (𝜑 → (dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆 ∩ 𝑇))
12		mpoeq12 7484	. . 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 595	. 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 49723	. . 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 595	. 2 ⊢ (𝜑 → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
18	13, 17	eqeltrrd 2870	1 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   ∩ cin 3912   × cxp 5660  dom cdm 5662   Fn wfn 6532  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  Subcatcsubc 17865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-pm 8826  df-ixp 8895  df-ssc 17866  df-subc 17868

This theorem is referenced by: (None)