Theorem mdvval 35536

Description: The set of disjoint variable conditions, which are pairs of distinct variables. (This definition differs from appendix C, which uses unordered pairs instead. We use ordered pairs, but all sets of disjoint variable conditions of interest will be symmetric, so it does not matter.) (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mdvval.v	⊢ 𝑉 = (mVR‘𝑇)
mdvval.d	⊢ 𝐷 = (mDV‘𝑇)

Assertion

Ref	Expression
mdvval	⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )

Proof of Theorem mdvval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdvval.d	. 2 ⊢ 𝐷 = (mDV‘𝑇)
2		fveq2 6822	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3		mdvval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
4	2, 3	eqtr4di 2784	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5	4	sqxpeqd 5648	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉))
6	5	difeq1d 4075	. . . 4 ⊢ (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I ))
7		df-mdv 35520	. . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8		fvex 6835	. . . . . 6 ⊢ (mVR‘𝑡) ∈ V
9	8, 8	xpex 7686	. . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10		difexg 5267	. . . . 5 ⊢ (((mVR‘𝑡) × (mVR‘𝑡)) ∈ V → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) ∈ V)
11	9, 10	ax-mp 5	. . . 4 ⊢ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) ∈ V
12	6, 7, 11	fvmpt3i 6934	. . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13		0dif 4355	. . . . 5 ⊢ (∅ ∖ I ) = ∅
14	13	eqcomi 2740	. . . 4 ⊢ ∅ = (∅ ∖ I )
15		fvprc 6814	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16		fvprc 6814	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2778	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	xpeq2d 5646	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19		xp0 6105	. . . . . 6 ⊢ (𝑉 × ∅) = ∅
20	18, 19	eqtrdi 2782	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
21	20	difeq1d 4075	. . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
22	14, 15, 21	3eqtr4a 2792	. . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
23	12, 22	pm2.61i 182	. 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
24	1, 23	eqtri 2754	1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3899  ∅c0 4283   I cid 5510   × cxp 5614  ‘cfv 6481  mVRcmvar 35493  mDVcmdv 35500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-mdv 35520

This theorem is referenced by:  mthmpps  35614  mclsppslem  35615