Theorem mdvval 35862

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 6871	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3		mdvval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
4	2, 3	eqtr4di 2818	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5	4	sqxpeqd 5683	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉))
6	5	difeq1d 4082	. . . 4 ⊢ (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I ))
7		df-mdv 35846	. . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8		fvex 6884	. . . . . 6 ⊢ (mVR‘𝑡) ∈ V
9	8, 8	xpex 7740	. . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10		difexg 5289	. . . . 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 6985	. . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13		0dif 4362	. . . . 5 ⊢ (∅ ∖ I ) = ∅
14	13	eqcomi 2774	. . . 4 ⊢ ∅ = (∅ ∖ I )
15		fvprc 6863	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16		fvprc 6863	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2812	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	xpeq2d 5681	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19		xp0 5751	. . . . . 6 ⊢ (𝑉 × ∅) = ∅
20	18, 19	eqtrdi 2816	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
21	20	difeq1d 4082	. . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
22	14, 15, 21	3eqtr4a 2826	. . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
23	12, 22	pm2.61i 184	. 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
24	1, 23	eqtri 2788	1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )

Syntax hints:  ¬ wn 3   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∖ cdif 3904  ∅c0 4288   I cid 5545   × cxp 5649  ‘cfv 6525  mVRcmvar 35819  mDVcmdv 35826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6481  df-fun 6527  df-fv 6533  df-mdv 35846

This theorem is referenced by:  mthmpps  35940  mclsppslem  35941