Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 15189 | . . . 4 ⊢ Rel ⇝ | |
2 | climuni 15249 | . . . . . . 7 ⊢ ((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) | |
3 | 2 | ax-gen 1798 | . . . . . 6 ⊢ ∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
4 | 3 | ax-gen 1798 | . . . . 5 ⊢ ∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
5 | 4 | ax-gen 1798 | . . . 4 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
6 | dffun2 6437 | . . . 4 ⊢ (Fun ⇝ ↔ (Rel ⇝ ∧ ∀𝑥∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧))) | |
7 | 1, 5, 6 | mpbir2an 708 | . . 3 ⊢ Fun ⇝ |
8 | funfn 6457 | . . 3 ⊢ (Fun ⇝ ↔ ⇝ Fn dom ⇝ ) | |
9 | 7, 8 | mpbi 229 | . 2 ⊢ ⇝ Fn dom ⇝ |
10 | vex 3434 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 10 | elrn 5796 | . . . 4 ⊢ (𝑦 ∈ ran ⇝ ↔ ∃𝑥 𝑥 ⇝ 𝑦) |
12 | climcl 15196 | . . . . 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 3925 | . 2 ⊢ ran ⇝ ⊆ ℂ |
16 | df-f 6431 | . 2 ⊢ ( ⇝ :dom ⇝ ⟶ℂ ↔ ( ⇝ Fn dom ⇝ ∧ ran ⇝ ⊆ ℂ)) | |
17 | 9, 15, 16 | mpbir2an 708 | 1 ⊢ ⇝ :dom ⇝ ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1537 ∃wex 1782 ∈ wcel 2106 ⊆ wss 3887 class class class wbr 5074 dom cdm 5585 ran crn 5586 Rel wrel 5590 Fun wfun 6421 Fn wfn 6422 ⟶wf 6423 ℂcc 10857 ⇝ cli 15181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-sup 9189 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-n0 12222 df-z 12308 df-uz 12571 df-rp 12719 df-seq 13710 df-exp 13771 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-clim 15185 |
This theorem is referenced by: climdm 15251 sum0 15421 sumz 15422 fsumsers 15428 isumclim 15457 isumcl 15461 ntrivcvgfvn0 15599 ntrivcvgtail 15600 zprodn0 15637 iprodclim 15696 iprodcl 15699 |
Copyright terms: Public domain | W3C validator |