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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 10783 | . . . 4 β’ (π΄ β π β (tarskiMapβπ΄) = β© {π₯ β Tarski β£ π΄ β π₯}) | |
2 | df-rab 3407 | . . . . 5 β’ {π₯ β Tarski β£ π΄ β π₯} = {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)} | |
3 | 2 | inteqi 4915 | . . . 4 β’ β© {π₯ β Tarski β£ π΄ β π₯} = β© {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)} |
4 | 1, 3 | eqtrdi 2789 | . . 3 β’ (π΄ β π β (tarskiMapβπ΄) = β© {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)}) |
5 | 4 | sseq2d 3980 | . 2 β’ (π΄ β π β (π΅ β (tarskiMapβπ΄) β π΅ β β© {π₯ β£ (π₯ β Tarski β§ π΄ β π₯)})) |
6 | impexp 452 | . . . 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 1822 | . . 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 4930 | . . 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 3062 | . . 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 397 βwal 1540 β wcel 2107 {cab 2710 βwral 3061 {crab 3406 β wss 3914 β© cint 4911 βcfv 6500 Tarskictsk 10692 tarskiMapctskm 10781 |
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-pr 5388 ax-groth 10767 |
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-int 4912 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-tsk 10693 df-tskm 10782 |
This theorem is referenced by: (None) |
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