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Mirrors > Home > ILE Home > Th. List > seqex | GIF version |
Description: Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqex | ⊢ seq𝑀( + , 𝐹) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-seqfrec 10219 | . 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
2 | frecex 6291 | . . 3 ⊢ frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) ∈ V | |
3 | 2 | rnex 4806 | . 2 ⊢ ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) ∈ V |
4 | 1, 3 | eqeltri 2212 | 1 ⊢ seq𝑀( + , 𝐹) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2686 〈cop 3530 ran crn 4540 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 freccfrec 6287 1c1 7621 + caddc 7623 ℤ≥cuz 9326 seqcseq 10218 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-recs 6202 df-frec 6288 df-seqfrec 10219 |
This theorem is referenced by: seq3shft 10610 clim2ser 11106 clim2ser2 11107 isermulc2 11109 iser3shft 11115 fsum3cvg 11147 sumrbdc 11148 isumclim3 11192 sumnul 11193 isumadd 11200 trireciplem 11269 geolim 11280 geolim2 11281 geo2lim 11285 geoisum1c 11289 mertensabs 11306 clim2prod 11308 clim2divap 11309 ntrivcvgap 11317 fproddccvg 11341 prodrbdclem2 11342 efcj 11379 eftlub 11396 eflegeo 11408 trilpolemisumle 13231 trilpolemeq1 13233 |
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