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Mirrors > Home > MPE Home > Th. List > sstskm | Structured version Visualization version GIF version |
Description: Being a part of (tarskiMapβπ΄). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
sstskm | β’ (π΄ β π β (π΅ β (tarskiMapβπ΄) β βπ₯ β Tarski (π΄ β π₯ β π΅ β π₯))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tskmval 10857 | . . . 4 β’ (π΄ β π β (tarskiMapβπ΄) = β© {π₯ β Tarski β£ π΄ β π₯}) | |
2 | df-rab 3429 | . . . . 5 β’ {π₯ β Tarski β£ π΄ β π₯} = {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)} | |
3 | 2 | inteqi 4949 | . . . 4 β’ β© {π₯ β Tarski β£ π΄ β π₯} = β© {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)} |
4 | 1, 3 | eqtrdi 2784 | . . 3 β’ (π΄ β π β (tarskiMapβπ΄) = β© {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)}) |
5 | 4 | sseq2d 4011 | . 2 β’ (π΄ β π β (π΅ β (tarskiMapβπ΄) β π΅ β β© {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)})) |
6 | impexp 450 | . . . 4 β’ (((π₯ β Tarski β§ π΄ β π₯) β π΅ β π₯) β (π₯ β Tarski β (π΄ β π₯ β π΅ β π₯))) | |
7 | 6 | albii 1814 | . . 3 β’ (βπ₯((π₯ β Tarski β§ π΄ β π₯) β π΅ β π₯) β βπ₯(π₯ β Tarski β (π΄ β π₯ β π΅ β π₯))) |
8 | ssintab 4964 | . . 3 β’ (π΅ β β© {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)} β βπ₯((π₯ β Tarski β§ π΄ β π₯) β π΅ β π₯)) | |
9 | df-ral 3058 | . . 3 β’ (βπ₯ β Tarski (π΄ β π₯ β π΅ β π₯) β βπ₯(π₯ β Tarski β (π΄ β π₯ β π΅ β π₯))) | |
10 | 7, 8, 9 | 3bitr4i 303 | . 2 β’ (π΅ β β© {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)} β βπ₯ β Tarski (π΄ β π₯ β π΅ β π₯)) |
11 | 5, 10 | bitrdi 287 | 1 β’ (π΄ β π β (π΅ β (tarskiMapβπ΄) β βπ₯ β Tarski (π΄ β π₯ β π΅ β π₯))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 βwal 1532 β wcel 2099 {cab 2705 βwral 3057 {crab 3428 β wss 3945 β© cint 4945 βcfv 6543 Tarskictsk 10766 tarskiMapctskm 10855 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-groth 10841 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-tsk 10767 df-tskm 10856 |
This theorem is referenced by: (None) |
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