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 dv 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 6191 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
3 | mdvval.v | . . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇) | |
4 | 2, 3 | syl6eqr 2674 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | 4 | sqxpeqd 5141 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mVR‘𝑡) × (mVR‘𝑡)) = (𝑉 × 𝑉)) |
6 | 5 | difeq1d 3727 | . . . 4 ⊢ (𝑡 = 𝑇 → (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ) = ((𝑉 × 𝑉) ∖ I )) |
7 | df-mdv 31385 | . . . 4 ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I )) | |
8 | fvex 6201 | . . . . . 6 ⊢ (mVR‘𝑡) ∈ V | |
9 | 8, 8 | xpex 6962 | . . . . 5 ⊢ ((mVR‘𝑡) × (mVR‘𝑡)) ∈ V |
10 | difexg 4808 | . . . . 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 6287 | . . 3 ⊢ (𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
13 | 0dif 3977 | . . . . 5 ⊢ (∅ ∖ I ) = ∅ | |
14 | 13 | eqcomi 2631 | . . . 4 ⊢ ∅ = (∅ ∖ I ) |
15 | fvprc 6185 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ∅) | |
16 | fvprc 6185 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
17 | 3, 16 | syl5eq 2668 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
18 | 17 | xpeq2d 5139 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = (𝑉 × ∅)) |
19 | xp0 5552 | . . . . . 6 ⊢ (𝑉 × ∅) = ∅ | |
20 | 18, 19 | syl6eq 2672 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑉 × 𝑉) = ∅) |
21 | 20 | difeq1d 3727 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ((𝑉 × 𝑉) ∖ I ) = (∅ ∖ I )) |
22 | 14, 15, 21 | 3eqtr4a 2682 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I )) |
23 | 12, 22 | pm2.61i 176 | . 2 ⊢ (mDV‘𝑇) = ((𝑉 × 𝑉) ∖ I ) |
24 | 1, 23 | eqtri 2644 | 1 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∅c0 3915 I cid 5023 × cxp 5112 ‘cfv 5888 mVRcmvar 31358 mDVcmdv 31365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-mdv 31385 |
This theorem is referenced by: mthmpps 31479 mclsppslem 31480 |
Copyright terms: Public domain | W3C validator |