Theorem hausflf 22848

Description: If a function has its values in a Hausdorff space, then it has at most one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
hausflf.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
hausflf	⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐿   𝑥,𝑋   𝑥,𝑌

Proof of Theorem hausflf

Step	Hyp	Ref	Expression
1		hausflimi 22831	. . 3 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
2	1	3ad2ant1 1135	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
3		haustop 22182	. . . . . 6 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
4		hausflf.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5	4	toptopon 21768	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	sylib 221	. . . . 5 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘𝑋))
7		flfval 22841	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
8	6, 7	syl3an1 1165	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
9	8	eleq2d 2816	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
10	9	mobidv 2548	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ ∃*𝑥 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
11	2, 10	mpbird 260	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))

Syntax hints:   → wi 4   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∃*wmo 2537  ∪ cuni 4805  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191  Topctop 21744  TopOnctopon 21761  Hauscha 22159  Filcfil 22696   FilMap cfm 22784   fLim cflim 22785   fLimf cflf 22786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-map 8488  df-fbas 20314  df-top 21745  df-topon 21762  df-nei 21949  df-haus 22166  df-fil 22697  df-flim 22790  df-flf 22791

This theorem is referenced by:  hausflf2  22849  cnextfun  22915  haustsms  22987  limcmo  24733