Theorem hausflf 23148

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 23131	. . 3 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
2	1	3ad2ant1 1132	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
3		haustop 22482	. . . . . 6 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
4		hausflf.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5	4	toptopon 22066	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	sylib 217	. . . . 5 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘𝑋))
7		flfval 23141	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
8	6, 7	syl3an1 1162	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
9	8	eleq2d 2824	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
10	9	mobidv 2549	. 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 256	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃*wmo 2538  ∪ cuni 4839  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Topctop 22042  TopOnctopon 22059  Hauscha 22459  Filcfil 22996   FilMap cfm 23084   fLim cflim 23085   fLimf cflf 23086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-fbas 20594  df-top 22043  df-topon 22060  df-nei 22249  df-haus 22466  df-fil 22997  df-flim 23090  df-flf 23091

This theorem is referenced by:  hausflf2  23149  cnextfun  23215  haustsms  23287  limcmo  25046