Theorem mopnval 22243

Description: An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object ( MetOpen ` D ) is the family of all open sets in the metric space determined by the metric D. By mopntop 22245, the open sets of a metric space form a topology J, whose base set is U. J by mopnuni 22246. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Hypothesis

Ref	Expression
mopnval.1	|- J = ( MetOpen ` D )

Assertion

Ref	Expression
mopnval	|- ( D e. ( *Met ` X ) -> J = ( topGen ` ran ( ball ` D ) ) )

Proof of Theorem mopnval

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvssunirn 6217	. . 3 |- ( *Met ` X ) C_ U. ran *Met
2	1	sseli 3599	. 2 |- ( D e. ( *Met ` X ) -> D e. U. ran *Met )
3		mopnval.1	. . 3 |- J = ( MetOpen ` D )
4		fveq2 6191	. . . . . 6 |- ( d = D -> ( ball ` d ) = ( ball ` D ) )
5	4	rneqd 5353	. . . . 5 |- ( d = D -> ran ( ball ` d ) = ran ( ball ` D ) )
6	5	fveq2d 6195	. . . 4 |- ( d = D -> ( topGen ` ran ( ball ` d ) ) = ( topGen ` ran ( ball ` D ) ) )
7		df-mopn 19742	. . . 4 |- MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
8		fvex 6201	. . . 4 |- ( topGen ` ran ( ball ` D ) ) e. _V
9	6, 7, 8	fvmpt 6282	. . 3 |- ( D e. U. ran *Met -> ( MetOpen ` D ) = ( topGen ` ran ( ball ` D ) ) )
10	3, 9	syl5eq 2668	. 2 |- ( D e. U. ran *Met -> J = ( topGen ` ran ( ball ` D ) ) )
11	2, 10	syl 17	1 |- ( D e. ( *Met ` X ) -> J = ( topGen ` ran ( ball ` D ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   U.cuni 4436   ran crn 5115   ` cfv 5888   topGenctg 16098   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-mopn 19742

This theorem is referenced by:  mopntopon  22244  elmopn  22247  imasf1oxms  22294  blssopn  22300  metss  22313  prdsxmslem2  22334  metcnp3  22345  xmetutop  22373  tgioo  22599  ismtyhmeolem  33603