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Theorem mdvval 31106
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 dv 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 6148 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mdvval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
42, 3syl6eqr 2673 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
54sqxpeqd 5101 . . . . 5 (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉))
65difeq1d 3705 . . . 4 (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I ))
7 df-mdv 31090 . . . 4 mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8 fvex 6158 . . . . . 6 (mVR‘𝑡) ∈ V
98, 8xpex 6915 . . . . 5 ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10 difexg 4768 . . . . 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 6244 . . 3 (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13 0dif 3949 . . . . 5 (∅ ∖ I ) = ∅
1413eqcomi 2630 . . . 4 ∅ = (∅ ∖ I )
15 fvprc 6142 . . . 4 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16 fvprc 6142 . . . . . . . 8 𝑇 ∈ V → (mVR‘𝑇) = ∅)
173, 16syl5eq 2667 . . . . . . 7 𝑇 ∈ V → 𝑉 = ∅)
1817xpeq2d 5099 . . . . . 6 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19 xp0 5511 . . . . . 6 (𝑉 × ∅) = ∅
2018, 19syl6eq 2671 . . . . 5 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
2120difeq1d 3705 . . . 4 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
2214, 15, 213eqtr4a 2681 . . 3 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
2312, 22pm2.61i 176 . 2 (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
241, 23eqtri 2643 1 𝐷 = ((𝑉 × 𝑉) ∖ I )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  c0 3891   I cid 4984   × cxp 5072  cfv 5847  mVRcmvar 31063  mDVcmdv 31070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-mdv 31090
This theorem is referenced by:  mthmpps  31184  mclsppslem  31185
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