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Mirrors > Home > MPE Home > Th. List > ttukeylem2 | Structured version Visualization version GIF version |
Description: Lemma for ttukey 10455. 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 486 | . . . . . 6 β’ ((π β§ π· β πΆ) β π· β πΆ) | |
2 | 1 | sspwd 4574 | . . . . 5 β’ ((π β§ π· β πΆ) β π« π· β π« πΆ) |
3 | ssrin 4194 | . . . . 5 β’ (π« π· β π« πΆ β (π« π· β© Fin) β (π« πΆ β© Fin)) | |
4 | sstr2 3952 | . . . . 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 10446 | . . . . 5 β’ (π β (πΆ β π΄ β (π« πΆ β© Fin) β π΄)) |
10 | 9 | adantr 482 | . . . 4 β’ ((π β§ π· β πΆ) β (πΆ β π΄ β (π« πΆ β© Fin) β π΄)) |
11 | 6, 7, 8 | ttukeylem1 10446 | . . . . 5 β’ (π β (π· β π΄ β (π« π· β© Fin) β π΄)) |
12 | 11 | adantr 482 | . . . 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 453 | . 2 β’ ((π β§ πΆ β π΄) β (π· β πΆ β π· β π΄)) |
15 | 14 | impr 456 | 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 397 βwal 1540 β wcel 2107 β cdif 3908 β© cin 3910 β wss 3911 π« cpw 4561 βͺ cuni 4866 β1-1-ontoβwf1o 6496 βcfv 6497 Fincfn 8884 cardccrd 9872 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-om 7804 df-1o 8413 df-en 8885 df-dom 8886 df-fin 8888 |
This theorem is referenced by: ttukeylem6 10451 ttukeylem7 10452 |
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