Theorem mdvval 34098

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 6842	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3		mdvval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
4	2, 3	eqtr4di 2794	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5	4	sqxpeqd 5665	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉))
6	5	difeq1d 4081	. . . 4 ⊢ (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I ))
7		df-mdv 34082	. . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8		fvex 6855	. . . . . 6 ⊢ (mVR‘𝑡) ∈ V
9	8, 8	xpex 7687	. . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10		difexg 5284	. . . . 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 6953	. . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13		0dif 4361	. . . . 5 ⊢ (∅ ∖ I ) = ∅
14	13	eqcomi 2745	. . . 4 ⊢ ∅ = (∅ ∖ I )
15		fvprc 6834	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16		fvprc 6834	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2788	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	xpeq2d 5663	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19		xp0 6110	. . . . . 6 ⊢ (𝑉 × ∅) = ∅
20	18, 19	eqtrdi 2792	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
21	20	difeq1d 4081	. . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
22	14, 15, 21	3eqtr4a 2802	. . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
23	12, 22	pm2.61i 182	. 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
24	1, 23	eqtri 2764	1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ∖ cdif 3907  ∅c0 4282   I cid 5530   × cxp 5631  ‘cfv 6496  mVRcmvar 34055  mDVcmdv 34062

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-mdv 34082

This theorem is referenced by:  mthmpps  34176  mclsppslem  34177