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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 14849 | . . . 4 ⊢ Rel ⇝ | |
2 | climuni 14909 | . . . . . . 7 ⊢ ((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) | |
3 | 2 | ax-gen 1796 | . . . . . 6 ⊢ ∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
4 | 3 | ax-gen 1796 | . . . . 5 ⊢ ∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
5 | 4 | ax-gen 1796 | . . . 4 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧) |
6 | dffun2 6365 | . . . 4 ⊢ (Fun ⇝ ↔ (Rel ⇝ ∧ ∀𝑥∀𝑦∀𝑧((𝑥 ⇝ 𝑦 ∧ 𝑥 ⇝ 𝑧) → 𝑦 = 𝑧))) | |
7 | 1, 5, 6 | mpbir2an 709 | . . 3 ⊢ Fun ⇝ |
8 | funfn 6385 | . . 3 ⊢ (Fun ⇝ ↔ ⇝ Fn dom ⇝ ) | |
9 | 7, 8 | mpbi 232 | . 2 ⊢ ⇝ Fn dom ⇝ |
10 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 10 | elrn 5822 | . . . 4 ⊢ (𝑦 ∈ ran ⇝ ↔ ∃𝑥 𝑥 ⇝ 𝑦) |
12 | climcl 14856 | . . . . 5 ⊢ (𝑥 ⇝ 𝑦 → 𝑦 ∈ ℂ) | |
13 | 12 | exlimiv 1931 | . . . 4 ⊢ (∃𝑥 𝑥 ⇝ 𝑦 → 𝑦 ∈ ℂ) |
14 | 11, 13 | sylbi 219 | . . 3 ⊢ (𝑦 ∈ ran ⇝ → 𝑦 ∈ ℂ) |
15 | 14 | ssriv 3971 | . 2 ⊢ ran ⇝ ⊆ ℂ |
16 | df-f 6359 | . 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 398 ∀wal 1535 ∃wex 1780 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ran crn 5556 Rel wrel 5560 Fun wfun 6349 Fn wfn 6350 ⟶wf 6351 ℂcc 10535 ⇝ cli 14841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 |
This theorem is referenced by: climdm 14911 sum0 15078 sumz 15079 fsumsers 15085 isumclim 15112 isumcl 15116 ntrivcvgfvn0 15255 ntrivcvgtail 15256 zprodn0 15293 iprodclim 15352 iprodcl 15355 |
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