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Mirrors > Home > MPE Home > Th. List > ackbij1lem17 | 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 |
---|---|
ackbij1lem17 | β’ πΉ:(π« Ο β© Fin)β1-1βΟ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ackbij.f | . . 3 β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) | |
2 | 1 | ackbij1lem10 10238 | . 2 β’ πΉ:(π« Ο β© Fin)βΆΟ |
3 | 1 | ackbij1lem16 10244 | . . 3 β’ ((π β (π« Ο β© Fin) β§ π β (π« Ο β© Fin)) β ((πΉβπ) = (πΉβπ) β π = π)) |
4 | 3 | rgen2 3192 | . 2 β’ βπ β (π« Ο β© Fin)βπ β (π« Ο β© Fin)((πΉβπ) = (πΉβπ) β π = π) |
5 | dff13 7259 | . 2 β’ (πΉ:(π« Ο β© Fin)β1-1βΟ β (πΉ:(π« Ο β© Fin)βΆΟ β§ βπ β (π« Ο β© Fin)βπ β (π« Ο β© Fin)((πΉβπ) = (πΉβπ) β π = π))) | |
6 | 2, 4, 5 | mpbir2an 710 | 1 β’ πΉ:(π« Ο β© Fin)β1-1βΟ |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 βwral 3056 β© cin 3943 π« cpw 4598 {csn 4624 βͺ ciun 4991 β¦ cmpt 5225 Γ cxp 5670 βΆwf 6538 β1-1βwf1 6539 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-pred 6299 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-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-dju 9910 df-card 9948 |
This theorem is referenced by: ackbij1 10247 ackbij1b 10248 ackbij2lem2 10249 |
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