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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 6907 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
3 | mdvval.v | . . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇) | |
4 | 2, 3 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | 4 | sqxpeqd 5721 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉)) |
6 | 5 | difeq1d 4135 | . . . 4 ⊢ (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I )) |
7 | df-mdv 35473 | . . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I )) | |
8 | fvex 6920 | . . . . . 6 ⊢ (mVR‘𝑡) ∈ V | |
9 | 8, 8 | xpex 7772 | . . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V |
10 | difexg 5335 | . . . . 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 7021 | . . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
13 | 0dif 4411 | . . . . 5 ⊢ (∅ ∖ I ) = ∅ | |
14 | 13 | eqcomi 2744 | . . . 4 ⊢ ∅ = (∅ ∖ I ) |
15 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅) | |
16 | fvprc 6899 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
17 | 3, 16 | eqtrid 2787 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
18 | 17 | xpeq2d 5719 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅)) |
19 | xp0 6180 | . . . . . 6 ⊢ (𝑉 × ∅) = ∅ | |
20 | 18, 19 | eqtrdi 2791 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅) |
21 | 20 | difeq1d 4135 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I )) |
22 | 14, 15, 21 | 3eqtr4a 2801 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
23 | 12, 22 | pm2.61i 182 | . 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ) |
24 | 1, 23 | eqtri 2763 | 1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ∅c0 4339 I cid 5582 × cxp 5687 ‘cfv 6563 mVRcmvar 35446 mDVcmdv 35453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-mdv 35473 |
This theorem is referenced by: mthmpps 35567 mclsppslem 35568 |
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