Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
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 2819 | . . 3 ⊢ (mDV‘𝑇) = (mDV‘𝑇) | |
2 | eqid 2819 | . . 3 ⊢ (mEx‘𝑇) = (mEx‘𝑇) | |
3 | mpstssv.p | . . 3 ⊢ 𝑃 = (mPreSt‘𝑇) | |
4 | 1, 2, 3 | mpstval 32775 | . 2 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) |
5 | xpss 5564 | . . 3 ⊢ ({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) ⊆ (V × V) | |
6 | ssv 3989 | . . 3 ⊢ (mEx‘𝑇) ⊆ V | |
7 | xpss12 5563 | . . 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 690 | . 2 ⊢ (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) ⊆ ((V × V) × V) |
9 | 4, 8 | eqsstri 3999 | 1 ⊢ 𝑃 ⊆ ((V × V) × V) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 {crab 3140 Vcvv 3493 ∩ cin 3933 ⊆ wss 3934 𝒫 cpw 4537 × cxp 5546 ◡ccnv 5547 ‘cfv 6348 Fincfn 8501 mExcmex 32707 mDVcmdv 32708 mPreStcmpst 32713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-mpst 32733 |
This theorem is referenced by: mpst123 32780 mpstrcl 32781 msrrcl 32783 elmpps 32813 |
Copyright terms: Public domain | W3C validator |