Theorem hausflf 24026

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 24009	. . 3 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
2	1	3ad2ant1 1133	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
3		haustop 23360	. . . . . 6 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
4		hausflf.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5	4	toptopon 22944	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	sylib 218	. . . . 5 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘𝑋))
7		flfval 24019	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
8	6, 7	syl3an1 1163	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
9	8	eleq2d 2830	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
10	9	mobidv 2552	. 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 257	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃*wmo 2541  ∪ cuni 4931  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Topctop 22920  TopOnctopon 22937  Hauscha 23337  Filcfil 23874   FilMap cfm 23962   fLim cflim 23963   fLimf cflf 23964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-fbas 21384  df-top 22921  df-topon 22938  df-nei 23127  df-haus 23344  df-fil 23875  df-flim 23968  df-flf 23969

This theorem is referenced by:  hausflf2  24027  cnextfun  24093  haustsms  24165  limcmo  25937