Theorem mdvval 36238

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 6877	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3		mdvval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
4	2, 3	eqtr4di 2814	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
5	4	sqxpeqd 5683	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉))
6	5	difeq1d 4073	. . . 4 ⊢ (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I ))
7		df-mdv 36222	. . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8		fvex 6890	. . . . . 6 ⊢ (mVR‘𝑡) ∈ V
9	8, 8	xpex 7756	. . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10		difexg 5291	. . . . 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 6991	. . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13		0dif 4356	. . . . 5 ⊢ (∅ ∖ I ) = ∅
14	13	eqcomi 2770	. . . 4 ⊢ ∅ = (∅ ∖ I )
15		fvprc 6869	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16		fvprc 6869	. . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅)
17	3, 16	eqtrid 2808	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅)
18	17	xpeq2d 5681	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19		xp0 5751	. . . . . 6 ⊢ (𝑉 × ∅) = ∅
20	18, 19	eqtrdi 2812	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
21	20	difeq1d 4073	. . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
22	14, 15, 21	3eqtr4a 2822	. . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
23	12, 22	pm2.61i 184	. 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
24	1, 23	eqtri 2784	1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896  ∅c0 4279   I cid 5545   × cxp 5649  ‘cfv 6531  mVRcmvar 36195  mDVcmdv 36202

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-mdv 36222

This theorem is used by:  mthmpps  36316  mclsppslem  36317