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Mirrors > Home > MPE Home > Th. List > tskmval | Structured version Visualization version GIF version |
Description: Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
tskmval | β’ (π΄ β π β (tarskiMapβπ΄) = β© {π₯ β Tarski β£ π΄ β π₯}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3466 | . 2 β’ (π΄ β π β π΄ β V) | |
2 | grothtsk 10778 | . . . . 5 β’ βͺ Tarski = V | |
3 | 1, 2 | eleqtrrdi 2849 | . . . 4 β’ (π΄ β π β π΄ β βͺ Tarski) |
4 | eluni2 4874 | . . . 4 β’ (π΄ β βͺ Tarski β βπ₯ β Tarski π΄ β π₯) | |
5 | 3, 4 | sylib 217 | . . 3 β’ (π΄ β π β βπ₯ β Tarski π΄ β π₯) |
6 | intexrab 5302 | . . 3 β’ (βπ₯ β Tarski π΄ β π₯ β β© {π₯ β Tarski β£ π΄ β π₯} β V) | |
7 | 5, 6 | sylib 217 | . 2 β’ (π΄ β π β β© {π₯ β Tarski β£ π΄ β π₯} β V) |
8 | eleq1 2826 | . . . . 5 β’ (π¦ = π΄ β (π¦ β π₯ β π΄ β π₯)) | |
9 | 8 | rabbidv 3418 | . . . 4 β’ (π¦ = π΄ β {π₯ β Tarski β£ π¦ β π₯} = {π₯ β Tarski β£ π΄ β π₯}) |
10 | 9 | inteqd 4917 | . . 3 β’ (π¦ = π΄ β β© {π₯ β Tarski β£ π¦ β π₯} = β© {π₯ β Tarski β£ π΄ β π₯}) |
11 | df-tskm 10781 | . . 3 β’ tarskiMap = (π¦ β V β¦ β© {π₯ β Tarski β£ π¦ β π₯}) | |
12 | 10, 11 | fvmptg 6951 | . 2 β’ ((π΄ β V β§ β© {π₯ β Tarski β£ π΄ β π₯} β V) β (tarskiMapβπ΄) = β© {π₯ β Tarski β£ π΄ β π₯}) |
13 | 1, 7, 12 | syl2anc 585 | 1 β’ (π΄ β π β (tarskiMapβπ΄) = β© {π₯ β Tarski β£ π΄ β π₯}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwrex 3074 {crab 3410 Vcvv 3448 βͺ cuni 4870 β© cint 4912 βcfv 6501 Tarskictsk 10691 tarskiMapctskm 10780 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-groth 10766 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-tsk 10692 df-tskm 10781 |
This theorem is referenced by: tskmid 10783 tskmcl 10784 sstskm 10785 eltskm 10786 |
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