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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 3487 | . 2 β’ (π΄ β π β π΄ β V) | |
2 | grothtsk 10829 | . . . . 5 β’ βͺ Tarski = V | |
3 | 1, 2 | eleqtrrdi 2838 | . . . 4 β’ (π΄ β π β π΄ β βͺ Tarski) |
4 | eluni2 4906 | . . . 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 5333 | . . 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 2815 | . . . . 5 β’ (π¦ = π΄ β (π¦ β π₯ β π΄ β π₯)) | |
9 | 8 | rabbidv 3434 | . . . 4 β’ (π¦ = π΄ β {π₯ β Tarski β£ π¦ β π₯} = {π₯ β Tarski β£ π΄ β π₯}) |
10 | 9 | inteqd 4948 | . . 3 β’ (π¦ = π΄ β β© {π₯ β Tarski β£ π¦ β π₯} = β© {π₯ β Tarski β£ π΄ β π₯}) |
11 | df-tskm 10832 | . . 3 β’ tarskiMap = (π¦ β V β¦ β© {π₯ β Tarski β£ π¦ β π₯}) | |
12 | 10, 11 | fvmptg 6989 | . 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 583 | 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 1533 β wcel 2098 βwrex 3064 {crab 3426 Vcvv 3468 βͺ cuni 4902 β© cint 4943 βcfv 6536 Tarskictsk 10742 tarskiMapctskm 10831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-groth 10817 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-tsk 10743 df-tskm 10832 |
This theorem is referenced by: tskmid 10834 tskmcl 10835 sstskm 10836 eltskm 10837 |
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