|   | Mathbox for Mario Carneiro | < Previous  
      Next > Nearby theorems | |
| 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 6906 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
| 3 | mdvval.v | . . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) | 
| 5 | 4 | sqxpeqd 5717 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉)) | 
| 6 | 5 | difeq1d 4125 | . . . 4 ⊢ (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I )) | 
| 7 | df-mdv 35493 | . . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I )) | |
| 8 | fvex 6919 | . . . . . 6 ⊢ (mVR‘𝑡) ∈ V | |
| 9 | 8, 8 | xpex 7773 | . . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V | 
| 10 | difexg 5329 | . . . . 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 4405 | . . . . 5 ⊢ (∅ ∖ I ) = ∅ | |
| 14 | 13 | eqcomi 2746 | . . . 4 ⊢ ∅ = (∅ ∖ I ) | 
| 15 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅) | |
| 16 | fvprc 6898 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
| 17 | 3, 16 | eqtrid 2789 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) | 
| 18 | 17 | xpeq2d 5715 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅)) | 
| 19 | xp0 6178 | . . . . . 6 ⊢ (𝑉 × ∅) = ∅ | |
| 20 | 18, 19 | eqtrdi 2793 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅) | 
| 21 | 20 | difeq1d 4125 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I )) | 
| 22 | 14, 15, 21 | 3eqtr4a 2803 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) | 
| 23 | 12, 22 | pm2.61i 182 | . 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ) | 
| 24 | 1, 23 | eqtri 2765 | 1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 I cid 5577 × cxp 5683 ‘cfv 6561 mVRcmvar 35466 mDVcmdv 35473 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-mdv 35493 | 
| This theorem is referenced by: mthmpps 35587 mclsppslem 35588 | 
| Copyright terms: Public domain | W3C validator |