Theorem hausflf2 22603

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:𝑌⟶𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1o)

Proof of Theorem hausflf2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4260	. . 3 ⊢ (((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
2	1	biimpi 219	. 2 ⊢ (((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅ → ∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
3		hausflf.x	. . 3 ⊢ 𝑋 = ∪ 𝐽
4	3	hausflf 22602	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
5		euen1b 8563	. . 3 ⊢ (((𝐽 fLimf 𝐿)‘𝐹) ≈ 1o ↔ ∃!𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
6		df-eu 2629	. . 3 ⊢ (∃!𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)))
7	5, 6	sylbbr 239	. 2 ⊢ ((∃𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1o)
8	2, 4, 7	syl2anr 599	1 ⊢ (((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1o)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃*wmo 2596  ∃!weu 2628   ≠ wne 2987  ∅c0 4243  ∪ cuni 4800   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  1_oc1o 8078   ≈ cen 8489  Hauscha 21913  Filcfil 22450   fLimf cflf 22540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1o 8085  df-map 8391  df-en 8493  df-fbas 20088  df-top 21499  df-topon 21516  df-nei 21703  df-haus 21920  df-fil 22451  df-flim 22544  df-flf 22545

This theorem is referenced by:  cnextfvval  22670  cnextcn  22672  cnextfres1  22673