Theorem hausflf 22605

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 22588	. . 3 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
2	1	3ad2ant1 1129	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
3		haustop 21939	. . . . . 6 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
4		hausflf.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5	4	toptopon 21525	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	sylib 220	. . . . 5 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘𝑋))
7		flfval 22598	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
8	6, 7	syl3an1 1159	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
9	8	eleq2d 2898	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝑥 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
10	9	mobidv 2633	. 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 259	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃*wmo 2620  ∪ cuni 4838  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  Topctop 21501  TopOnctopon 21518  Hauscha 21916  Filcfil 22453   FilMap cfm 22541   fLim cflim 22542   fLimf cflf 22543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408  df-fbas 20542  df-top 21502  df-topon 21519  df-nei 21706  df-haus 21923  df-fil 22454  df-flim 22547  df-flf 22548

This theorem is referenced by:  hausflf2  22606  cnextfun  22672  haustsms  22744  limcmo  24480