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Mirrors > Home > MPE Home > Th. List > fclim | Structured version Visualization version GIF version |
Description: The limit relation is function-like, and with codomain the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
fclim | ⊢ ⇝ :dom ⇝ ⟶ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrel 15366 | . . . 4 ⊢ Rel ⇝ | |
2 | climuni 15426 | . . . . . . 7 ⊢ ((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) | |
3 | 2 | ax-gen 1797 | . . . . . 6 ⊢ ∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
4 | 3 | ax-gen 1797 | . . . . 5 ⊢ ∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
5 | 4 | ax-gen 1797 | . . . 4 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
6 | dffun2 6503 | . . . 4 ⊢ (Fun ⇝ ↔ (Rel ⇝ ∧ ∀𝑥∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧))) | |
7 | 1, 5, 6 | mpbir2an 709 | . . 3 ⊢ Fun ⇝ |
8 | funfn 6528 | . . 3 ⊢ (Fun ⇝ ↔ ⇝ Fn dom ⇝ ) | |
9 | 7, 8 | mpbi 229 | . 2 ⊢ ⇝ Fn dom ⇝ |
10 | vex 3447 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 10 | elrn 5847 | . . . 4 ⊢ (𝑦 ∈ ran ⇝ ↔ ∃𝑥 𝑥 ⇝ 𝑦) |
12 | climcl 15373 | . . . . 5 ⊢ (𝑥 ⇝ 𝑦 → 𝑦 ∈ ℂ) | |
13 | 12 | exlimiv 1933 | . . . 4 ⊢ (∃𝑥 𝑥 ⇝ 𝑦 → 𝑦 ∈ ℂ) |
14 | 11, 13 | sylbi 216 | . . 3 ⊢ (𝑦 ∈ ran ⇝ → 𝑦 ∈ ℂ) |
15 | 14 | ssriv 3946 | . 2 ⊢ ran ⇝ ⊆ ℂ |
16 | df-f 6497 | . 2 ⊢ ( ⇝ :dom ⇝ ⟶ℂ ↔ ( ⇝ Fn dom ⇝ ∧ ran ⇝ ⊆ ℂ)) | |
17 | 9, 15, 16 | mpbir2an 709 | 1 ⊢ ⇝ :dom ⇝ ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1539 ∃wex 1781 ∈ wcel 2106 ⊆ wss 3908 class class class wbr 5103 dom cdm 5631 ran crn 5632 Rel wrel 5636 Fun wfun 6487 Fn wfn 6488 ⟶wf 6489 ℂcc 11045 ⇝ cli 15358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 df-seq 13899 df-exp 13960 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-clim 15362 |
This theorem is referenced by: climdm 15428 sum0 15598 sumz 15599 fsumsers 15605 isumclim 15634 isumcl 15638 ntrivcvgfvn0 15776 ntrivcvgtail 15777 zprodn0 15814 iprodclim 15873 iprodcl 15876 |
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