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Theorem mdvval 34162
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 6846 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mdvval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
42, 3eqtr4di 2791 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
54sqxpeqd 5669 . . . . 5 (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉))
65difeq1d 4085 . . . 4 (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I ))
7 df-mdv 34146 . . . 4 mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8 fvex 6859 . . . . . 6 (mVR‘𝑡) ∈ V
98, 8xpex 7691 . . . . 5 ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10 difexg 5288 . . . . 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 6957 . . 3 (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13 0dif 4365 . . . . 5 (∅ ∖ I ) = ∅
1413eqcomi 2742 . . . 4 ∅ = (∅ ∖ I )
15 fvprc 6838 . . . 4 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16 fvprc 6838 . . . . . . . 8 𝑇 ∈ V → (mVR‘𝑇) = ∅)
173, 16eqtrid 2785 . . . . . . 7 𝑇 ∈ V → 𝑉 = ∅)
1817xpeq2d 5667 . . . . . 6 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19 xp0 6114 . . . . . 6 (𝑉 × ∅) = ∅
2018, 19eqtrdi 2789 . . . . 5 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
2120difeq1d 4085 . . . 4 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
2214, 15, 213eqtr4a 2799 . . 3 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
2312, 22pm2.61i 182 . 2 (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
241, 23eqtri 2761 1 𝐷 = ((𝑉 × 𝑉) ∖ I )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3447  cdif 3911  c0 4286   I cid 5534   × cxp 5635  cfv 6500  mVRcmvar 34119  mDVcmdv 34126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-mdv 34146
This theorem is referenced by:  mthmpps  34240  mclsppslem  34241
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