Theorem flimtopon 21774

Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
flimtopon	|- ( A e. ( J fLim F ) -> ( J e. (TopOn ` X ) <-> F e. ( Fil ` X ) ) )

Proof of Theorem flimtopon

Step	Hyp	Ref	Expression
1		flimtop 21769	. . 3 |- ( A e. ( J fLim F ) -> J e. Top )
2		istopon 20717	. . . 4 |- ( J e. (TopOn ` X ) <-> ( J e. Top /\ X = U. J ) )
3	2	baib 944	. . 3 |- ( J e. Top -> ( J e. (TopOn ` X ) <-> X = U. J ) )
4	1, 3	syl 17	. 2 |- ( A e. ( J fLim F ) -> ( J e. (TopOn ` X ) <-> X = U. J ) )
5		eqid 2622	. . . . 5 |- U. J = U. J
6	5	flimfil 21773	. . . 4 |- ( A e. ( J fLim F ) -> F e. ( Fil ` U. J ) )
7		fveq2 6191	. . . . 5 |- ( X = U. J -> ( Fil ` X ) = ( Fil ` U. J ) )
8	7	eleq2d 2687	. . . 4 |- ( X = U. J -> ( F e. ( Fil ` X ) <-> F e. ( Fil ` U. J ) ) )
9	6, 8	syl5ibrcom 237	. . 3 |- ( A e. ( J fLim F ) -> ( X = U. J -> F e. ( Fil ` X ) ) )
10		filunibas 21685	. . . . 5 |- ( F e. ( Fil ` U. J ) -> U. F = U. J )
11	6, 10	syl 17	. . . 4 |- ( A e. ( J fLim F ) -> U. F = U. J )
12		filunibas 21685	. . . . 5 |- ( F e. ( Fil ` X ) -> U. F = X )
13	12	eqeq1d 2624	. . . 4 |- ( F e. ( Fil ` X ) -> ( U. F = U. J <-> X = U. J ) )
14	11, 13	syl5ibcom 235	. . 3 |- ( A e. ( J fLim F ) -> ( F e. ( Fil ` X ) -> X = U. J ) )
15	9, 14	impbid 202	. 2 |- ( A e. ( J fLim F ) -> ( X = U. J <-> F e. ( Fil ` X ) ) )
16	4, 15	bitrd 268	1 |- ( A e. ( J fLim F ) -> ( J e. (TopOn ` X ) <-> F e. ( Fil ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   U.cuni 4436   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   Filcfil 21649   fLim cflim 21738

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-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-fbas 19743  df-top 20699  df-topon 20716  df-nei 20902  df-fil 21650  df-flim 21743

This theorem is referenced by:  fclsfnflim  21831