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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 range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
fclim | ⊢ ⇝ :dom ⇝ ⟶ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrel 14599 | . . . 4 ⊢ Rel ⇝ | |
2 | climuni 14659 | . . . . . . 7 ⊢ ((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) | |
3 | 2 | ax-gen 1896 | . . . . . 6 ⊢ ∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
4 | 3 | ax-gen 1896 | . . . . 5 ⊢ ∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
5 | 4 | ax-gen 1896 | . . . 4 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
6 | dffun2 6132 | . . . 4 ⊢ (Fun ⇝ ↔ (Rel ⇝ ∧ ∀𝑥∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧))) | |
7 | 1, 5, 6 | mpbir2an 704 | . . 3 ⊢ Fun ⇝ |
8 | funfn 6152 | . . 3 ⊢ (Fun ⇝ ↔ ⇝ Fn dom ⇝ ) | |
9 | 7, 8 | mpbi 222 | . 2 ⊢ ⇝ Fn dom ⇝ |
10 | vex 3416 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 10 | elrn 5598 | . . . 4 ⊢ (𝑦 ∈ ran ⇝ ↔ ∃𝑥 𝑥 ⇝ 𝑦) |
12 | climcl 14606 | . . . . 5 ⊢ (𝑥 ⇝ 𝑦 → 𝑦 ∈ ℂ) | |
13 | 12 | exlimiv 2031 | . . . 4 ⊢ (∃𝑥 𝑥 ⇝ 𝑦 → 𝑦 ∈ ℂ) |
14 | 11, 13 | sylbi 209 | . . 3 ⊢ (𝑦 ∈ ran ⇝ → 𝑦 ∈ ℂ) |
15 | 14 | ssriv 3830 | . 2 ⊢ ran ⇝ ⊆ ℂ |
16 | df-f 6126 | . 2 ⊢ ( ⇝ :dom ⇝ ⟶ℂ ↔ ( ⇝ Fn dom ⇝ ∧ ran ⇝ ⊆ ℂ)) | |
17 | 9, 15, 16 | mpbir2an 704 | 1 ⊢ ⇝ :dom ⇝ ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∀wal 1656 ∃wex 1880 ∈ wcel 2166 ⊆ wss 3797 class class class wbr 4872 dom cdm 5341 ran crn 5342 Rel wrel 5346 Fun wfun 6116 Fn wfn 6117 ⟶wf 6118 ℂcc 10249 ⇝ cli 14591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-sup 8616 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-rp 12112 df-seq 13095 df-exp 13154 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-clim 14595 |
This theorem is referenced by: climdm 14661 sum0 14828 sumz 14829 fsumsers 14835 isumclim 14862 isumcl 14866 ntrivcvgfvn0 15003 ntrivcvgtail 15004 zprodn0 15041 iprodclim 15100 iprodcl 15103 |
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