Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  infsubc2d Structured version   Visualization version   GIF version

Theorem infsubc2d 49680
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 6626 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
32dmeqd 5881 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
4 dmxpid 5906 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
53, 4eqtrdi 2813 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
6 infsubc2d.2 . . . . . . 7 (𝜑𝐽 Fn (𝑇 × 𝑇))
76fndmd 6626 . . . . . 6 (𝜑 → dom 𝐽 = (𝑇 × 𝑇))
87dmeqd 5881 . . . . 5 (𝜑 → dom dom 𝐽 = dom (𝑇 × 𝑇))
9 dmxpid 5906 . . . . 5 dom (𝑇 × 𝑇) = 𝑇
108, 9eqtrdi 2813 . . . 4 (𝜑 → dom dom 𝐽 = 𝑇)
115, 10ineq12d 4173 . . 3 (𝜑 → (dom dom 𝐻 ∩ dom dom 𝐽) = (𝑆𝑇))
12 mpoeq12 7469 . . 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 593 . 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 49679 . . 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 593 . 2 (𝜑 → (𝑥 ∈ (dom dom 𝐻 ∩ dom dom 𝐽), 𝑦 ∈ (dom dom 𝐻 ∩ dom dom 𝐽) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
1813, 17eqeltrrd 2863 1 (𝜑 → (𝑥 ∈ (𝑆𝑇), 𝑦 ∈ (𝑆𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1560  wcel 2142  cin 3903   × cxp 5645  dom cdm 5647   Fn wfn 6516  cfv 6521  (class class class)co 7396  cmpo 7398  Subcatcsubc 17842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-pm 8811  df-ixp 8880  df-ssc 17843  df-subc 17845
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator