Theorem infsubc2d 49549

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 6597	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
3	2	dmeqd 5854	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
4		dmxpid 5879	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
5	3, 4	eqtrdi 2788	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
6		infsubc2d.2	. . . . . . 7 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
7	6	fndmd 6597	. . . . . 6 ⊢ (𝜑 → dom 𝐽 = (𝑇 × 𝑇))
8	7	dmeqd 5854	. . . . 5 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑇 × 𝑇))
9		dmxpid 5879	. . . . 5 ⊢ dom (𝑇 × 𝑇) = 𝑇
10	8, 9	eqtrdi 2788	. . . 4 ⊢ (𝜑 → dom dom 𝐽 = 𝑇)
11	5, 10	ineq12d 4162	. . 3 ⊢ (𝜑 → (dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆 ∩ 𝑇))
12		mpoeq12 7433	. . 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 585	. 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 49548	. . 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 585	. 2 ⊢ (𝜑 → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
18	13, 17	eqeltrrd 2838	1 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∩ cin 3889   × cxp 5622  dom cdm 5624   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  Subcatcsubc 17767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-pm 8769  df-ixp 8839  df-ssc 17768  df-subc 17770

This theorem is referenced by: (None)