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Mirrors > Home > MPE Home > Th. List > ackbij1lem10 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 10232. (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 4196 | . . . 4 β’ (π₯ β (π« Ο β© Fin) β π₯ β Fin) | |
3 | snfi 9043 | . . . . . 6 β’ {π¦} β Fin | |
4 | elinel1 4195 | . . . . . . . . . 10 β’ (π₯ β (π« Ο β© Fin) β π₯ β π« Ο) | |
5 | 4 | elpwid 4611 | . . . . . . . . 9 β’ (π₯ β (π« Ο β© Fin) β π₯ β Ο) |
6 | onfin2 9230 | . . . . . . . . . 10 β’ Ο = (On β© Fin) | |
7 | inss2 4229 | . . . . . . . . . 10 β’ (On β© Fin) β Fin | |
8 | 6, 7 | eqsstri 4016 | . . . . . . . . 9 β’ Ο β Fin |
9 | 5, 8 | sstrdi 3994 | . . . . . . . 8 β’ (π₯ β (π« Ο β© Fin) β π₯ β Fin) |
10 | 9 | sselda 3982 | . . . . . . 7 β’ ((π₯ β (π« Ο β© Fin) β§ π¦ β π₯) β π¦ β Fin) |
11 | pwfi 9177 | . . . . . . 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 9316 | . . . . . 6 β’ (({π¦} β Fin β§ π« π¦ β Fin) β ({π¦} Γ π« π¦) β Fin) | |
14 | 3, 12, 13 | sylancr 587 | . . . . 5 β’ ((π₯ β (π« Ο β© Fin) β§ π¦ β π₯) β ({π¦} Γ π« π¦) β Fin) |
15 | 14 | ralrimiva 3146 | . . . 4 β’ (π₯ β (π« Ο β© Fin) β βπ¦ β π₯ ({π¦} Γ π« π¦) β Fin) |
16 | iunfi 9339 | . . . 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 584 | . . 3 β’ (π₯ β (π« Ο β© Fin) β βͺ π¦ β π₯ ({π¦} Γ π« π¦) β Fin) |
18 | ficardom 9955 | . . 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 7111 | 1 β’ πΉ:(π« Ο β© Fin)βΆΟ |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 β© cin 3947 π« cpw 4602 {csn 4628 βͺ ciun 4997 β¦ cmpt 5231 Γ cxp 5674 Oncon0 6364 βΆwf 6539 βcfv 6543 Οcom 7854 Fincfn 8938 cardccrd 9929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7855 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 |
This theorem is referenced by: ackbij1lem12 10225 ackbij1lem13 10226 ackbij1lem14 10227 ackbij1lem15 10228 ackbij1lem16 10229 ackbij1lem17 10230 ackbij1lem18 10231 ackbij1b 10233 |
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