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Mirrors > Home > MPE Home > Th. List > isf32lem12 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-2 10364. (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 8845 | . . . . 5 β’ (π β (π« πΊ βm Ο) β π:ΟβΆπ« πΊ) | |
2 | isf32lem11 10360 | . . . . . . . . . 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 413 | . . . . . . . . 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 451 | . . . . . . 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 419 | . . . . 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 3146 | . 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 10326 | . 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 259 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 {cab 2703 βwral 3055 β wss 3943 π« cpw 4597 β© cint 4943 class class class wbr 5141 ran crn 5670 suc csuc 6360 βΆwf 6533 βcfv 6537 (class class class)co 7405 Οcom 7852 βm cmap 8822 βΌ* cwdom 9561 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-wdom 9562 df-card 9936 |
This theorem is referenced by: isf33lem 10363 isfin3-2 10364 |
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