Theorem infsubc2d 49024

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 6605	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
3	2	dmeqd 5859	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
4		dmxpid 5883	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
5	3, 4	eqtrdi 2780	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
6		infsubc2d.2	. . . . . . 7 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
7	6	fndmd 6605	. . . . . 6 ⊢ (𝜑 → dom 𝐽 = (𝑇 × 𝑇))
8	7	dmeqd 5859	. . . . 5 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑇 × 𝑇))
9		dmxpid 5883	. . . . 5 ⊢ dom (𝑇 × 𝑇) = 𝑇
10	8, 9	eqtrdi 2780	. . . 4 ⊢ (𝜑 → dom dom 𝐽 = 𝑇)
11	5, 10	ineq12d 4180	. . 3 ⊢ (𝜑 → (dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆 ∩ 𝑇))
12		mpoeq12 7442	. . 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 584	. 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 49023	. . 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 584	. 2 ⊢ (𝜑 → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
18	13, 17	eqeltrrd 2829	1 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∩ cin 3910   × cxp 5629  dom cdm 5631   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  Subcatcsubc 17747

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-pm 8779  df-ixp 8848  df-ssc 17748  df-subc 17750

This theorem is referenced by: (None)