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Mathbox for Mario Carneiro |
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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 | β’ π = (mPreStβπ) |
Ref | Expression |
---|---|
mpstssv | β’ π β ((V Γ V) Γ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (mDVβπ) = (mDVβπ) | |
2 | eqid 2733 | . . 3 β’ (mExβπ) = (mExβπ) | |
3 | mpstssv.p | . . 3 β’ π = (mPreStβπ) | |
4 | 1, 2, 3 | mpstval 34193 | . 2 β’ π = (({π β π« (mDVβπ) β£ β‘π = π} Γ (π« (mExβπ) β© Fin)) Γ (mExβπ)) |
5 | xpss 5653 | . . 3 β’ ({π β π« (mDVβπ) β£ β‘π = π} Γ (π« (mExβπ) β© Fin)) β (V Γ V) | |
6 | ssv 3972 | . . 3 β’ (mExβπ) β V | |
7 | xpss12 5652 | . . 3 β’ ((({π β π« (mDVβπ) β£ β‘π = π} Γ (π« (mExβπ) β© Fin)) β (V Γ V) β§ (mExβπ) β V) β (({π β π« (mDVβπ) β£ β‘π = π} Γ (π« (mExβπ) β© Fin)) Γ (mExβπ)) β ((V Γ V) Γ V)) | |
8 | 5, 6, 7 | mp2an 691 | . 2 β’ (({π β π« (mDVβπ) β£ β‘π = π} Γ (π« (mExβπ) β© Fin)) Γ (mExβπ)) β ((V Γ V) Γ V) |
9 | 4, 8 | eqsstri 3982 | 1 β’ π β ((V Γ V) Γ V) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {crab 3406 Vcvv 3447 β© cin 3913 β wss 3914 π« cpw 4564 Γ cxp 5635 β‘ccnv 5636 βcfv 6500 Fincfn 8889 mExcmex 34125 mDVcmdv 34126 mPreStcmpst 34131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-mpst 34151 |
This theorem is referenced by: mpst123 34198 mpstrcl 34199 msrrcl 34201 elmpps 34231 |
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