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Mirrors > Home > MPE Home > Th. List > ackbij1lem10 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 10247. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
Ref | Expression |
---|---|
ackbij.f | β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) |
Ref | Expression |
---|---|
ackbij1lem10 | β’ πΉ:(π« Ο β© Fin)βΆΟ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ackbij.f | . 2 β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) | |
2 | elinel2 4192 | . . . 4 β’ (π₯ β (π« Ο β© Fin) β π₯ β Fin) | |
3 | snfi 9058 | . . . . . 6 β’ {π¦} β Fin | |
4 | elinel1 4191 | . . . . . . . . . 10 β’ (π₯ β (π« Ο β© Fin) β π₯ β π« Ο) | |
5 | 4 | elpwid 4607 | . . . . . . . . 9 β’ (π₯ β (π« Ο β© Fin) β π₯ β Ο) |
6 | onfin2 9245 | . . . . . . . . . 10 β’ Ο = (On β© Fin) | |
7 | inss2 4225 | . . . . . . . . . 10 β’ (On β© Fin) β Fin | |
8 | 6, 7 | eqsstri 4012 | . . . . . . . . 9 β’ Ο β Fin |
9 | 5, 8 | sstrdi 3990 | . . . . . . . 8 β’ (π₯ β (π« Ο β© Fin) β π₯ β Fin) |
10 | 9 | sselda 3978 | . . . . . . 7 β’ ((π₯ β (π« Ο β© Fin) β§ π¦ β π₯) β π¦ β Fin) |
11 | pwfi 9192 | . . . . . . 7 β’ (π¦ β Fin β π« π¦ β Fin) | |
12 | 10, 11 | sylib 217 | . . . . . 6 β’ ((π₯ β (π« Ο β© Fin) β§ π¦ β π₯) β π« π¦ β Fin) |
13 | xpfi 9331 | . . . . . 6 β’ (({π¦} β Fin β§ π« π¦ β Fin) β ({π¦} Γ π« π¦) β Fin) | |
14 | 3, 12, 13 | sylancr 586 | . . . . 5 β’ ((π₯ β (π« Ο β© Fin) β§ π¦ β π₯) β ({π¦} Γ π« π¦) β Fin) |
15 | 14 | ralrimiva 3141 | . . . 4 β’ (π₯ β (π« Ο β© Fin) β βπ¦ β π₯ ({π¦} Γ π« π¦) β Fin) |
16 | iunfi 9354 | . . . 4 β’ ((π₯ β Fin β§ βπ¦ β π₯ ({π¦} Γ π« π¦) β Fin) β βͺ π¦ β π₯ ({π¦} Γ π« π¦) β Fin) | |
17 | 2, 15, 16 | syl2anc 583 | . . 3 β’ (π₯ β (π« Ο β© Fin) β βͺ π¦ β π₯ ({π¦} Γ π« π¦) β Fin) |
18 | ficardom 9970 | . . 3 β’ (βͺ π¦ β π₯ ({π¦} Γ π« π¦) β Fin β (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦)) β Ο) | |
19 | 17, 18 | syl 17 | . 2 β’ (π₯ β (π« Ο β© Fin) β (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦)) β Ο) |
20 | 1, 19 | fmpti 7116 | 1 β’ πΉ:(π« Ο β© Fin)βΆΟ |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1534 β wcel 2099 βwral 3056 β© cin 3943 π« cpw 4598 {csn 4624 βͺ ciun 4991 β¦ cmpt 5225 Γ cxp 5670 Oncon0 6363 βΆwf 6538 βcfv 6542 Οcom 7862 Fincfn 8953 cardccrd 9944 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7863 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-card 9948 |
This theorem is referenced by: ackbij1lem12 10240 ackbij1lem13 10241 ackbij1lem14 10242 ackbij1lem15 10243 ackbij1lem16 10244 ackbij1lem17 10245 ackbij1lem18 10246 ackbij1b 10248 |
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