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Mirrors > Home > MPE Home > Th. List > ttukeylem2 | Structured version Visualization version GIF version |
Description: Lemma for ttukey 10515. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
ttukeylem.1 | β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) |
ttukeylem.2 | β’ (π β π΅ β π΄) |
ttukeylem.3 | β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) |
Ref | Expression |
---|---|
ttukeylem2 | β’ ((π β§ (πΆ β π΄ β§ π· β πΆ)) β π· β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . . 6 β’ ((π β§ π· β πΆ) β π· β πΆ) | |
2 | 1 | sspwd 4610 | . . . . 5 β’ ((π β§ π· β πΆ) β π« π· β π« πΆ) |
3 | ssrin 4228 | . . . . 5 β’ (π« π· β π« πΆ β (π« π· β© Fin) β (π« πΆ β© Fin)) | |
4 | sstr2 3984 | . . . . 5 β’ ((π« π· β© Fin) β (π« πΆ β© Fin) β ((π« πΆ β© Fin) β π΄ β (π« π· β© Fin) β π΄)) | |
5 | 2, 3, 4 | 3syl 18 | . . . 4 β’ ((π β§ π· β πΆ) β ((π« πΆ β© Fin) β π΄ β (π« π· β© Fin) β π΄)) |
6 | ttukeylem.1 | . . . . . 6 β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) | |
7 | ttukeylem.2 | . . . . . 6 β’ (π β π΅ β π΄) | |
8 | ttukeylem.3 | . . . . . 6 β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) | |
9 | 6, 7, 8 | ttukeylem1 10506 | . . . . 5 β’ (π β (πΆ β π΄ β (π« πΆ β© Fin) β π΄)) |
10 | 9 | adantr 480 | . . . 4 β’ ((π β§ π· β πΆ) β (πΆ β π΄ β (π« πΆ β© Fin) β π΄)) |
11 | 6, 7, 8 | ttukeylem1 10506 | . . . . 5 β’ (π β (π· β π΄ β (π« π· β© Fin) β π΄)) |
12 | 11 | adantr 480 | . . . 4 β’ ((π β§ π· β πΆ) β (π· β π΄ β (π« π· β© Fin) β π΄)) |
13 | 5, 10, 12 | 3imtr4d 294 | . . 3 β’ ((π β§ π· β πΆ) β (πΆ β π΄ β π· β π΄)) |
14 | 13 | impancom 451 | . 2 β’ ((π β§ πΆ β π΄) β (π· β πΆ β π· β π΄)) |
15 | 14 | impr 454 | 1 β’ ((π β§ (πΆ β π΄ β§ π· β πΆ)) β π· β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 βwal 1531 β wcel 2098 β cdif 3940 β© cin 3942 β wss 3943 π« cpw 4597 βͺ cuni 4902 β1-1-ontoβwf1o 6536 βcfv 6537 Fincfn 8941 cardccrd 9932 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-1o 8467 df-en 8942 df-dom 8943 df-fin 8945 |
This theorem is referenced by: ttukeylem6 10511 ttukeylem7 10512 |
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