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Mirrors > Home > MPE Home > Th. List > Mathboxes > mdvval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
mdvval.v | ⊢ 𝑉 = (mVR‘𝑇) |
mdvval.d | ⊢ 𝐷 = (mDV‘𝑇) |
Ref | Expression |
---|---|
mdvval | ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I ) |
Step | Hyp | Ref | 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) | |
4 | 2, 3 | eqtr4di 2790 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | 4 | sqxpeqd 5708 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉)) |
6 | 5 | difeq1d 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 | |
9 | 8, 8 | xpex 7739 | . . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V |
10 | difexg 5327 | . . . . 5 ⊢ (((mVR‘𝑡) × (mVR‘𝑡)) ∈ V → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) ∈ V) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) ∈ V |
12 | 6, 7, 11 | fvmpt3i 7003 | . . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
13 | 0dif 4401 | . . . . 5 ⊢ (∅ ∖ I ) = ∅ | |
14 | 13 | eqcomi 2741 | . . . 4 ⊢ ∅ = (∅ ∖ I ) |
15 | fvprc 6883 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅) | |
16 | fvprc 6883 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
17 | 3, 16 | eqtrid 2784 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
18 | 17 | xpeq2d 5706 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅)) |
19 | xp0 6157 | . . . . . 6 ⊢ (𝑉 × ∅) = ∅ | |
20 | 18, 19 | eqtrdi 2788 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅) |
21 | 20 | difeq1d 4121 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I )) |
22 | 14, 15, 21 | 3eqtr4a 2798 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
23 | 12, 22 | pm2.61i 182 | . 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ) |
24 | 1, 23 | eqtri 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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