![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isf32lem12 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-2 10388. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
isf32lem40.f | β’ πΉ = {π β£ βπ β (π« π βm Ο)(βπ₯ β Ο (πβsuc π₯) β (πβπ₯) β β© ran π β ran π)} |
Ref | Expression |
---|---|
isf32lem12 | β’ (πΊ β π β (Β¬ Ο βΌ* πΊ β πΊ β πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8864 | . . . . 5 β’ (π β (π« πΊ βm Ο) β π:ΟβΆπ« πΊ) | |
2 | isf32lem11 10384 | . . . . . . . . . 10 β’ ((πΊ β π β§ (π:ΟβΆπ« πΊ β§ βπ β Ο (πβsuc π) β (πβπ) β§ Β¬ β© ran π β ran π)) β Ο βΌ* πΊ) | |
3 | 2 | expcom 412 | . . . . . . . . 9 β’ ((π:ΟβΆπ« πΊ β§ βπ β Ο (πβsuc π) β (πβπ) β§ Β¬ β© ran π β ran π) β (πΊ β π β Ο βΌ* πΊ)) |
4 | 3 | 3expa 1115 | . . . . . . . 8 β’ (((π:ΟβΆπ« πΊ β§ βπ β Ο (πβsuc π) β (πβπ)) β§ Β¬ β© ran π β ran π) β (πΊ β π β Ο βΌ* πΊ)) |
5 | 4 | impancom 450 | . . . . . . 7 β’ (((π:ΟβΆπ« πΊ β§ βπ β Ο (πβsuc π) β (πβπ)) β§ πΊ β π) β (Β¬ β© ran π β ran π β Ο βΌ* πΊ)) |
6 | 5 | con1d 145 | . . . . . 6 β’ (((π:ΟβΆπ« πΊ β§ βπ β Ο (πβsuc π) β (πβπ)) β§ πΊ β π) β (Β¬ Ο βΌ* πΊ β β© ran π β ran π)) |
7 | 6 | exp31 418 | . . . . 5 β’ (π:ΟβΆπ« πΊ β (βπ β Ο (πβsuc π) β (πβπ) β (πΊ β π β (Β¬ Ο βΌ* πΊ β β© ran π β ran π)))) |
8 | 1, 7 | syl 17 | . . . 4 β’ (π β (π« πΊ βm Ο) β (βπ β Ο (πβsuc π) β (πβπ) β (πΊ β π β (Β¬ Ο βΌ* πΊ β β© ran π β ran π)))) |
9 | 8 | com4t 93 | . . 3 β’ (πΊ β π β (Β¬ Ο βΌ* πΊ β (π β (π« πΊ βm Ο) β (βπ β Ο (πβsuc π) β (πβπ) β β© ran π β ran π)))) |
10 | 9 | ralrimdv 3142 | . 2 β’ (πΊ β π β (Β¬ Ο βΌ* πΊ β βπ β (π« πΊ βm Ο)(βπ β Ο (πβsuc π) β (πβπ) β β© ran π β ran π))) |
11 | isf32lem40.f | . . 3 β’ πΉ = {π β£ βπ β (π« π βm Ο)(βπ₯ β Ο (πβsuc π₯) β (πβπ₯) β β© ran π β ran π)} | |
12 | 11 | isfin3ds 10350 | . 2 β’ (πΊ β π β (πΊ β πΉ β βπ β (π« πΊ βm Ο)(βπ β Ο (πβsuc π) β (πβπ) β β© ran π β ran π))) |
13 | 10, 12 | sylibrd 258 | 1 β’ (πΊ β π β (Β¬ Ο βΌ* πΊ β πΊ β πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 {cab 2702 βwral 3051 β wss 3940 π« cpw 4598 β© cint 4944 class class class wbr 5143 ran crn 5673 suc csuc 6366 βΆwf 6538 βcfv 6542 (class class class)co 7415 Οcom 7867 βm cmap 8841 βΌ* cwdom 9585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-wdom 9586 df-card 9960 |
This theorem is referenced by: isf33lem 10387 isfin3-2 10388 |
Copyright terms: Public domain | W3C validator |