Theorem hausflf2 21802

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 = U. J

Assertion

Ref	Expression
hausflf2	|- ( ( ( J e. Haus /\ L e. ( 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 3931	. . 3 |- ( ( ( J fLimf L ) ` F ) =/= (/) <-> E. x x e. ( ( J fLimf L ) ` F ) )
2	1	biimpi 206	. 2 |- ( ( ( J fLimf L ) ` F ) =/= (/) -> E. x x e. ( ( J fLimf L ) ` F ) )
3		hausflf.x	. . 3 |- X = U. J
4	3	hausflf 21801	. 2 |- ( ( J e. Haus /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> E* x x e. ( ( J fLimf L ) ` F ) )
5		euen1b 8027	. . . 4 |- ( ( ( J fLimf L ) ` F ) ~~ 1o <-> E! x x e. ( ( J fLimf L ) ` F ) )
6		eu5 2496	. . . 4 |- ( E! x x e. ( ( J fLimf L ) ` F ) <-> ( E. x x e. ( ( J fLimf L ) ` F ) /\ E* x x e. ( ( J fLimf L ) ` F ) ) )
7	5, 6	bitr2i 265	. . 3 |- ( ( E. x x e. ( ( J fLimf L ) ` F ) /\ E* x x e. ( ( J fLimf L ) ` F ) ) <-> ( ( J fLimf L ) ` F ) ~~ 1o )
8	7	biimpi 206	. 2 |- ( ( E. x x e. ( ( J fLimf L ) ` F ) /\ E* x x e. ( ( J fLimf L ) ` F ) ) -> ( ( J fLimf L ) ` F ) ~~ 1o )
9	2, 4, 8	syl2anr 495	1 |- ( ( ( J e. Haus /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ ( ( J fLimf L ) ` F ) =/= (/) ) -> ( ( J fLimf L ) ` F ) ~~ 1o )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   =/= wne 2794   (/)c0 3915   U.cuni 4436   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ~~ cen 7952   Hauscha 21112   Filcfil 21649   fLimf cflf 21739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-map 7859  df-en 7956  df-fbas 19743  df-top 20699  df-topon 20716  df-nei 20902  df-haus 21119  df-fil 21650  df-flim 21743  df-flf 21744

This theorem is referenced by:  cnextfvval  21869  cnextcn  21871  cnextfres1  21872