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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 6726 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
3 | mdvval.v | . . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇) | |
4 | 2, 3 | eqtr4di 2797 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | 4 | sqxpeqd 5592 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉)) |
6 | 5 | difeq1d 4045 | . . . 4 ⊢ (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I )) |
7 | df-mdv 33176 | . . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I )) | |
8 | fvex 6739 | . . . . . 6 ⊢ (mVR‘𝑡) ∈ V | |
9 | 8, 8 | xpex 7547 | . . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V |
10 | difexg 5229 | . . . . 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 6832 | . . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
13 | 0dif 4325 | . . . . 5 ⊢ (∅ ∖ I ) = ∅ | |
14 | 13 | eqcomi 2747 | . . . 4 ⊢ ∅ = (∅ ∖ I ) |
15 | fvprc 6718 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅) | |
16 | fvprc 6718 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
17 | 3, 16 | syl5eq 2791 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
18 | 17 | xpeq2d 5590 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅)) |
19 | xp0 6030 | . . . . . 6 ⊢ (𝑉 × ∅) = ∅ | |
20 | 18, 19 | eqtrdi 2795 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅) |
21 | 20 | difeq1d 4045 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I )) |
22 | 14, 15, 21 | 3eqtr4a 2805 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
23 | 12, 22 | pm2.61i 185 | . 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ) |
24 | 1, 23 | eqtri 2766 | 1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2111 Vcvv 3415 ∖ cdif 3872 ∅c0 4246 I cid 5463 × cxp 5558 ‘cfv 6389 mVRcmvar 33149 mDVcmdv 33156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3417 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-iota 6347 df-fun 6391 df-fv 6397 df-mdv 33176 |
This theorem is referenced by: mthmpps 33270 mclsppslem 33271 |
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