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Theorem infsubc2d 49631
Description: The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.)
Hypotheses
Ref Expression
infsubc2d.1 (𝜑𝐻 Fn (𝑆 × 𝑆))
infsubc2d.2 (𝜑𝐽 Fn (𝑇 × 𝑇))
infsubc2d.3 (𝜑𝐻 ∈ (Subcat‘𝐶))
infsubc2d.4 (𝜑𝐽 ∈ (Subcat‘𝐶))
Assertion
Ref Expression
infsubc2d (𝜑 → (𝑥 ∈ (𝑆𝑇), 𝑦 ∈ (𝑆𝑇) ↦ ((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
StepHypRef Expression
1 infsubc2d.1 . . . . . . 7 (𝜑𝐻 Fn (𝑆 × 𝑆))
21fndmd 6615 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
32dmeqd 5874 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
4 dmxpid 5899 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
53, 4eqtrdi 2807 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
6 infsubc2d.2 . . . . . . 7 (𝜑𝐽 Fn (𝑇 × 𝑇))
76fndmd 6615 . . . . . 6 (𝜑 → dom 𝐽 = (𝑇 × 𝑇))
87dmeqd 5874 . . . . 5 (𝜑 → dom dom 𝐽 = dom (𝑇 × 𝑇))
9 dmxpid 5899 . . . . 5 dom (𝑇 × 𝑇) = 𝑇
108, 9eqtrdi 2807 . . . 4 (𝜑 → dom dom 𝐽 = 𝑇)
115, 10ineq12d 4168 . . 3 (𝜑 → (dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆𝑇))
12 mpoeq12 7458 . . 3 (((dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆𝑇) ∧ (dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆𝑇)) → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) = (𝑥 ∈ (𝑆𝑇), 𝑦 ∈ (𝑆𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))))
1311, 11, 12syl2anc 592 . 2 (𝜑 → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) = (𝑥 ∈ (𝑆𝑇), 𝑦 ∈ (𝑆𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))))
14 infsubc2d.3 . . 3 (𝜑𝐻 ∈ (Subcat‘𝐶))
15 infsubc2d.4 . . 3 (𝜑𝐽 ∈ (Subcat‘𝐶))
16 infsubc2 49630 . . 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))
1714, 15, 16syl2anc 592 . 2 (𝜑 → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
1813, 17eqeltrrd 2857 1 (𝜑 → (𝑥 ∈ (𝑆𝑇), 𝑦 ∈ (𝑆𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1554  wcel 2136  cin 3898   × cxp 5638  dom cdm 5640   Fn wfn 6505  cfv 6510  (class class class)co 7385  cmpo 7387  Subcatcsubc 17818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-iin 4946  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-1st 7959  df-2nd 7960  df-pm 8799  df-ixp 8869  df-ssc 17819  df-subc 17821
This theorem is referenced by: (None)
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