Theorem infsubc2d 50315

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 6652	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
3	2	dmeqd 5891	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
4		dmxpid 5916	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
5	3, 4	eqtrdi 2814	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
6		infsubc2d.2	. . . . . . 7 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
7	6	fndmd 6652	. . . . . 6 ⊢ (𝜑 → dom 𝐽 = (𝑇 × 𝑇))
8	7	dmeqd 5891	. . . . 5 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑇 × 𝑇))
9		dmxpid 5916	. . . . 5 ⊢ dom (𝑇 × 𝑇) = 𝑇
10	8, 9	eqtrdi 2814	. . . 4 ⊢ (𝜑 → dom dom 𝐽 = 𝑇)
11	5, 10	ineq12d 4120	. . 3 ⊢ (𝜑 → (dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆 ∩ 𝑇))
12		mpoeq12 7500	. . 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 50314	. . 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 2864	1 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146   ∩ cin 4091   × cxp 5652  dom cdm 5654   Fn wfn 6541  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429  Subcatcsubc 17996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-iin 4956  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-pm 8861  df-ixp 8937  df-ssc 17997  df-subc 17999

This theorem is used by: (None)