Theorem mdvval 35977

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 6883	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3		mdvval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
4	2, 3	eqtr4di 2816	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5	4	sqxpeqd 5695	. . . . 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 35961	. . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8		fvex 6896	. . . . . 6 ⊢ (mVR‘𝑡) ∈ V
9	8, 8	xpex 7753	. . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10		difexg 5301	. . . . 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 6997	. . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13		0dif 4364	. . . . 5 ⊢ (∅ ∖ I ) = ∅
14	13	eqcomi 2772	. . . 4 ⊢ ∅ = (∅ ∖ I )
15		fvprc 6875	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16		fvprc 6875	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2810	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	xpeq2d 5693	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19		xp0 5763	. . . . . 6 ⊢ (𝑉 × ∅) = ∅
20	18, 19	eqtrdi 2814	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
21	20	difeq1d 4081	. . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
22	14, 15, 21	3eqtr4a 2824	. . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
23	12, 22	pm2.61i 184	. 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
24	1, 23	eqtri 2786	1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3903  ∅c0 4287   I cid 5557   × cxp 5661  ‘cfv 6538  mVRcmvar 35934  mDVcmdv 35941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-mdv 35961

This theorem is referenced by:  mthmpps  36055  mclsppslem  36056