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Theorem mdvval 34490
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.
StepHypRef Expression
1 mdvval.d . 2 𝐷 = (mDV‘𝑇)
2 fveq2 6891 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mdvval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
42, 3eqtr4di 2790 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
54sqxpeqd 5708 . . . . 5 (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉))
65difeq1d 4121 . . . 4 (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I ))
7 df-mdv 34474 . . . 4 mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8 fvex 6904 . . . . . 6 (mVR‘𝑡) ∈ V
98, 8xpex 7739 . . . . 5 ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10 difexg 5327 . . . . 5 (((mVR‘𝑡) × (mVR‘𝑡)) ∈ V → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) ∈ V)
119, 10ax-mp 5 . . . 4 (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) ∈ V
126, 7, 11fvmpt3i 7003 . . 3 (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13 0dif 4401 . . . . 5 (∅ ∖ I ) = ∅
1413eqcomi 2741 . . . 4 ∅ = (∅ ∖ I )
15 fvprc 6883 . . . 4 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16 fvprc 6883 . . . . . . . 8 𝑇 ∈ V → (mVR‘𝑇) = ∅)
173, 16eqtrid 2784 . . . . . . 7 𝑇 ∈ V → 𝑉 = ∅)
1817xpeq2d 5706 . . . . . 6 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19 xp0 6157 . . . . . 6 (𝑉 × ∅) = ∅
2018, 19eqtrdi 2788 . . . . 5 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
2120difeq1d 4121 . . . 4 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
2214, 15, 213eqtr4a 2798 . . 3 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
2312, 22pm2.61i 182 . 2 (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
241, 23eqtri 2760 1 𝐷 = ((𝑉 × 𝑉) ∖ I )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106  Vcvv 3474  cdif 3945  c0 4322   I cid 5573   × cxp 5674  cfv 6543  mVRcmvar 34447  mDVcmdv 34454
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-mdv 34474
This theorem is referenced by:  mthmpps  34568  mclsppslem  34569
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