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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpstssv | Structured version Visualization version GIF version |
Description: A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mpstssv.p | ⊢ 𝑃 = (mPreSt‘𝑇) |
Ref | Expression |
---|---|
mpstssv | ⊢ 𝑃 ⊆ ((V × V) × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (mDV‘𝑇) = (mDV‘𝑇) | |
2 | eqid 2610 | . . 3 ⊢ (mEx‘𝑇) = (mEx‘𝑇) | |
3 | mpstssv.p | . . 3 ⊢ 𝑃 = (mPreSt‘𝑇) | |
4 | 1, 2, 3 | mpstval 30686 | . 2 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) |
5 | xpss 5149 | . . 3 ⊢ ({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) ⊆ (V × V) | |
6 | ssv 3588 | . . 3 ⊢ (mEx‘𝑇) ⊆ V | |
7 | xpss12 5148 | . . 3 ⊢ ((({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) ⊆ (V × V) ∧ (mEx‘𝑇) ⊆ V) → (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) ⊆ ((V × V) × V)) | |
8 | 5, 6, 7 | mp2an 704 | . 2 ⊢ (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) ⊆ ((V × V) × V) |
9 | 4, 8 | eqsstri 3598 | 1 ⊢ 𝑃 ⊆ ((V × V) × V) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 × cxp 5036 ◡ccnv 5037 ‘cfv 5804 Fincfn 7841 mExcmex 30618 mDVcmdv 30619 mPreStcmpst 30624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-mpst 30644 |
This theorem is referenced by: mpst123 30691 mpstrcl 30692 msrrcl 30694 elmpps 30724 |
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