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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 10445 | . 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) | |
2 | frecex 6394 | . . 3 ⊢ frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ∈ V | |
3 | 2 | rnex 4894 | . 2 ⊢ ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ∈ V |
4 | 1, 3 | eqeltri 2250 | 1 ⊢ seq𝑀( + , 𝐹) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 Vcvv 2737 ⟨cop 3595 ran crn 4627 ‘cfv 5216 (class class class)co 5874 ∈ cmpo 5876 freccfrec 6390 1c1 7811 + caddc 7813 ℤ≥cuz 9527 seqcseq 10444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-recs 6305 df-frec 6391 df-seqfrec 10445 |
This theorem is referenced by: seq3shft 10846 clim2ser 11344 clim2ser2 11345 isermulc2 11347 iser3shft 11353 fsum3cvg 11385 sumrbdc 11386 isumclim3 11430 sumnul 11431 isumadd 11438 trireciplem 11507 geolim 11518 geolim2 11519 geo2lim 11523 geoisum1c 11527 mertensabs 11544 clim2prod 11546 clim2divap 11547 ntrivcvgap 11555 fproddccvg 11579 prodrbdclem2 11580 fprodntrivap 11591 efcj 11680 eftlub 11697 eflegeo 11708 nninfdc 12453 mulgfvalg 12984 trilpolemisumle 14756 trilpolemeq1 14758 |
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