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Theorem hausflf2 22003
 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.)
Hypothesis
Ref Expression
hausflf.x 𝑋 = 𝐽
Assertion
Ref Expression
hausflf2 (((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜)

Proof of Theorem hausflf2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 n0 4074 . . 3 (((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
21biimpi 206 . 2 (((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅ → ∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
3 hausflf.x . . 3 𝑋 = 𝐽
43hausflf 22002 . 2 ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
5 euen1b 8192 . . . 4 (((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜 ↔ ∃!𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
6 eu5 2633 . . . 4 (∃!𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)))
75, 6bitr2i 265 . . 3 ((∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) ↔ ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜)
87biimpi 206 . 2 ((∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜)
92, 4, 8syl2anr 496 1 (((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1𝑜)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632  ∃wex 1853   ∈ wcel 2139  ∃!weu 2607  ∃*wmo 2608   ≠ wne 2932  ∅c0 4058  ∪ cuni 4588   class class class wbr 4804  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813  1𝑜c1o 7722   ≈ cen 8118  Hauscha 21314  Filcfil 21850   fLimf cflf 21940 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1o 7729  df-map 8025  df-en 8122  df-fbas 19945  df-top 20901  df-topon 20918  df-nei 21104  df-haus 21321  df-fil 21851  df-flim 21944  df-flf 21945 This theorem is referenced by:  cnextfvval  22070  cnextcn  22072  cnextfres1  22073
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