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Mirrors > Home > MPE Home > Th. List > ackbij1lem10 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 10182. (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 4160 | . . . 4 β’ (π₯ β (π« Ο β© Fin) β π₯ β Fin) | |
3 | snfi 8994 | . . . . . 6 β’ {π¦} β Fin | |
4 | elinel1 4159 | . . . . . . . . . 10 β’ (π₯ β (π« Ο β© Fin) β π₯ β π« Ο) | |
5 | 4 | elpwid 4573 | . . . . . . . . 9 β’ (π₯ β (π« Ο β© Fin) β π₯ β Ο) |
6 | onfin2 9181 | . . . . . . . . . 10 β’ Ο = (On β© Fin) | |
7 | inss2 4193 | . . . . . . . . . 10 β’ (On β© Fin) β Fin | |
8 | 6, 7 | eqsstri 3982 | . . . . . . . . 9 β’ Ο β Fin |
9 | 5, 8 | sstrdi 3960 | . . . . . . . 8 β’ (π₯ β (π« Ο β© Fin) β π₯ β Fin) |
10 | 9 | sselda 3948 | . . . . . . 7 β’ ((π₯ β (π« Ο β© Fin) β§ π¦ β π₯) β π¦ β Fin) |
11 | pwfi 9128 | . . . . . . 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 9267 | . . . . . 6 β’ (({π¦} β Fin β§ π« π¦ β Fin) β ({π¦} Γ π« π¦) β Fin) | |
14 | 3, 12, 13 | sylancr 588 | . . . . 5 β’ ((π₯ β (π« Ο β© Fin) β§ π¦ β π₯) β ({π¦} Γ π« π¦) β Fin) |
15 | 14 | ralrimiva 3140 | . . . 4 β’ (π₯ β (π« Ο β© Fin) β βπ¦ β π₯ ({π¦} Γ π« π¦) β Fin) |
16 | iunfi 9290 | . . . 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 585 | . . 3 β’ (π₯ β (π« Ο β© Fin) β βͺ π¦ β π₯ ({π¦} Γ π« π¦) β Fin) |
18 | ficardom 9905 | . . 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 7064 | 1 β’ πΉ:(π« Ο β© Fin)βΆΟ |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 β© cin 3913 π« cpw 4564 {csn 4590 βͺ ciun 4958 β¦ cmpt 5192 Γ cxp 5635 Oncon0 6321 βΆwf 6496 βcfv 6500 Οcom 7806 Fincfn 8889 cardccrd 9879 |
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-pow 5324 ax-pr 5388 ax-un 7676 |
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 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-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 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-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-om 7807 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 |
This theorem is referenced by: ackbij1lem12 10175 ackbij1lem13 10176 ackbij1lem14 10177 ackbij1lem15 10178 ackbij1lem16 10179 ackbij1lem17 10180 ackbij1lem18 10181 ackbij1b 10183 |
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