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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 2798 | . . 3 ⊢ (mDV‘𝑇) = (mDV‘𝑇) | |
2 | eqid 2798 | . . 3 ⊢ (mEx‘𝑇) = (mEx‘𝑇) | |
3 | mpstssv.p | . . 3 ⊢ 𝑃 = (mPreSt‘𝑇) | |
4 | 1, 2, 3 | mpstval 32895 | . 2 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) |
5 | xpss 5535 | . . 3 ⊢ ({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) ⊆ (V × V) | |
6 | ssv 3939 | . . 3 ⊢ (mEx‘𝑇) ⊆ V | |
7 | xpss12 5534 | . . 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 691 | . 2 ⊢ (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) ⊆ ((V × V) × V) |
9 | 4, 8 | eqsstri 3949 | 1 ⊢ 𝑃 ⊆ ((V × V) × V) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 {crab 3110 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 × cxp 5517 ◡ccnv 5518 ‘cfv 6324 Fincfn 8492 mExcmex 32827 mDVcmdv 32828 mPreStcmpst 32833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-mpst 32853 |
This theorem is referenced by: mpst123 32900 mpstrcl 32901 msrrcl 32903 elmpps 32933 |
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