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Mirrors > Home > MPE Home > Th. List > hausflf2 | Structured version Visualization version GIF version |
Description: If a convergent function has its values in a Hausdorff space, then it has a unique limit. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
Ref | Expression |
---|---|
hausflf.x | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
hausflf2 | ⊢ (((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3931 | . . 3 ⊢ (((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) | |
2 | 1 | biimpi 206 | . 2 ⊢ (((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅ → ∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) |
3 | hausflf.x | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | hausflf 21801 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) |
5 | euen1b 8027 | . . . 4 ⊢ (((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜 ↔ ∃!𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) | |
6 | eu5 2496 | . . . 4 ⊢ (∃!𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))) | |
7 | 5, 6 | bitr2i 265 | . . 3 ⊢ ((∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) ↔ ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜) |
8 | 7 | biimpi 206 | . 2 ⊢ ((∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜) |
9 | 2, 4, 8 | syl2anr 495 | 1 ⊢ (((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 ∃*wmo 2471 ≠ wne 2794 ∅c0 3915 ∪ cuni 4436 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 ≈ cen 7952 Hauscha 21112 Filcfil 21649 fLimf cflf 21739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1o 7560 df-map 7859 df-en 7956 df-fbas 19743 df-top 20699 df-topon 20716 df-nei 20902 df-haus 21119 df-fil 21650 df-flim 21743 df-flf 21744 |
This theorem is referenced by: cnextfvval 21869 cnextcn 21871 cnextfres1 21872 |
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