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Mirrors > Home > ILE Home > Th. List > seqex | Unicode version |
Description: Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-seqfrec 10395 | . 2 frec | |
2 | frecex 6371 | . . 3 frec | |
3 | 2 | rnex 4876 | . 2 frec |
4 | 1, 3 | eqeltri 2243 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 2141 cvv 2730 cop 3584 crn 4610 cfv 5196 (class class class)co 5851 cmpo 5853 freccfrec 6367 c1 7768 caddc 7770 cuz 9480 cseq 10394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-recs 6282 df-frec 6368 df-seqfrec 10395 |
This theorem is referenced by: seq3shft 10795 clim2ser 11293 clim2ser2 11294 isermulc2 11296 iser3shft 11302 fsum3cvg 11334 sumrbdc 11335 isumclim3 11379 sumnul 11380 isumadd 11387 trireciplem 11456 geolim 11467 geolim2 11468 geo2lim 11472 geoisum1c 11476 mertensabs 11493 clim2prod 11495 clim2divap 11496 ntrivcvgap 11504 fproddccvg 11528 prodrbdclem2 11529 fprodntrivap 11540 efcj 11629 eftlub 11646 eflegeo 11657 nninfdc 12401 trilpolemisumle 14035 trilpolemeq1 14037 |
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