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Theorem mdvval 32744
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 6663 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
3 mdvval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
42, 3syl6eqr 2872 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
54sqxpeqd 5580 . . . . 5 (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉))
65difeq1d 4096 . . . 4 (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I ))
7 df-mdv 32728 . . . 4 mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
8 fvex 6676 . . . . . 6 (mVR‘𝑡) ∈ V
98, 8xpex 7468 . . . . 5 ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V
10 difexg 5222 . . . . 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 6766 . . 3 (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
13 0dif 4353 . . . . 5 (∅ ∖ I ) = ∅
1413eqcomi 2828 . . . 4 ∅ = (∅ ∖ I )
15 fvprc 6656 . . . 4 𝑇 ∈ V → (mDV‘𝑇) = ∅)
16 fvprc 6656 . . . . . . . 8 𝑇 ∈ V → (mVR‘𝑇) = ∅)
173, 16syl5eq 2866 . . . . . . 7 𝑇 ∈ V → 𝑉 = ∅)
1817xpeq2d 5578 . . . . . 6 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅))
19 xp0 6008 . . . . . 6 (𝑉 × ∅) = ∅
2018, 19syl6eq 2870 . . . . 5 𝑇 ∈ V → (𝑉 × 𝑉) = ∅)
2120difeq1d 4096 . . . 4 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I ))
2214, 15, 213eqtr4a 2880 . . 3 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ))
2312, 22pm2.61i 184 . 2 (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )
241, 23eqtri 2842 1 𝐷 = ((𝑉 × 𝑉) ∖ I )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1531  wcel 2108  Vcvv 3493  cdif 3931  c0 4289   I cid 5452   × cxp 5546  cfv 6348  mVRcmvar 32701  mDVcmdv 32708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-mdv 32728
This theorem is referenced by:  mthmpps  32822  mclsppslem  32823
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