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Theorem mdvval 31733
 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 6332 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mdvval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
42, 3syl6eqr 2822 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
54sqxpeqd 5281 . . . . 5 (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉))
65difeq1d 3876 . . . 4 (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I ))
7 df-mdv 31717 . . . 4 mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8 fvex 6342 . . . . . 6 (mVR‘𝑡) ∈ V
98, 8xpex 7108 . . . . 5 ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10 difexg 4939 . . . . 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 6429 . . 3 (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13 0dif 4119 . . . . 5 (∅ ∖ I ) = ∅
1413eqcomi 2779 . . . 4 ∅ = (∅ ∖ I )
15 fvprc 6326 . . . 4 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16 fvprc 6326 . . . . . . . 8 𝑇 ∈ V → (mVR‘𝑇) = ∅)
173, 16syl5eq 2816 . . . . . . 7 𝑇 ∈ V → 𝑉 = ∅)
1817xpeq2d 5279 . . . . . 6 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19 xp0 5693 . . . . . 6 (𝑉 × ∅) = ∅
2018, 19syl6eq 2820 . . . . 5 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
2120difeq1d 3876 . . . 4 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
2214, 15, 213eqtr4a 2830 . . 3 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
2312, 22pm2.61i 176 . 2 (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
241, 23eqtri 2792 1 𝐷 = ((𝑉 × 𝑉) ∖ I )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1630   ∈ wcel 2144  Vcvv 3349   ∖ cdif 3718  ∅c0 4061   I cid 5156   × cxp 5247  ‘cfv 6031  mVRcmvar 31690  mDVcmdv 31697 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-mdv 31717 This theorem is referenced by:  mthmpps  31811  mclsppslem  31812
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