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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 6846 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
3 | mdvval.v | . . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇) | |
4 | 2, 3 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | 4 | sqxpeqd 5669 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉)) |
6 | 5 | difeq1d 4085 | . . . 4 ⊢ (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I )) |
7 | df-mdv 34146 | . . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I )) | |
8 | fvex 6859 | . . . . . 6 ⊢ (mVR‘𝑡) ∈ V | |
9 | 8, 8 | xpex 7691 | . . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V |
10 | difexg 5288 | . . . . 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 6957 | . . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
13 | 0dif 4365 | . . . . 5 ⊢ (∅ ∖ I ) = ∅ | |
14 | 13 | eqcomi 2742 | . . . 4 ⊢ ∅ = (∅ ∖ I ) |
15 | fvprc 6838 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅) | |
16 | fvprc 6838 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
17 | 3, 16 | eqtrid 2785 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
18 | 17 | xpeq2d 5667 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅)) |
19 | xp0 6114 | . . . . . 6 ⊢ (𝑉 × ∅) = ∅ | |
20 | 18, 19 | eqtrdi 2789 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅) |
21 | 20 | difeq1d 4085 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I )) |
22 | 14, 15, 21 | 3eqtr4a 2799 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
23 | 12, 22 | pm2.61i 182 | . 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ) |
24 | 1, 23 | eqtri 2761 | 1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∖ cdif 3911 ∅c0 4286 I cid 5534 × cxp 5635 ‘cfv 6500 mVRcmvar 34119 mDVcmdv 34126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-mdv 34146 |
This theorem is referenced by: mthmpps 34240 mclsppslem 34241 |
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