Theorem pnrmcld 21146

Description: A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
pnrmcld	|- ( ( J e. PNrm /\ A e. ( Clsd ` J ) ) -> E. f e. ( J ^m NN ) A = |^| ran f )

Distinct variable groups:   A, f   f, J

Proof of Theorem pnrmcld

Step	Hyp	Ref	Expression
1		ispnrm 21143	. . . 4 |- ( J e. PNrm <-> ( J e. Nrm /\ ( Clsd ` J ) C_ ran ( f e. ( J ^m NN ) |-> |^| ran f ) ) )
2	1	simprbi 480	. . 3 |- ( J e. PNrm -> ( Clsd ` J ) C_ ran ( f e. ( J ^m NN ) |-> |^| ran f ) )
3	2	sselda 3603	. 2 |- ( ( J e. PNrm /\ A e. ( Clsd ` J ) ) -> A e. ran ( f e. ( J ^m NN ) |-> |^| ran f ) )
4		eqid 2622	. . . 4 |- ( f e. ( J ^m NN ) |-> |^| ran f ) = ( f e. ( J ^m NN ) |-> |^| ran f )
5	4	elrnmpt 5372	. . 3 |- ( A e. ( Clsd ` J ) -> ( A e. ran ( f e. ( J ^m NN ) |-> |^| ran f ) <-> E. f e. ( J ^m NN ) A = |^| ran f ) )
6	5	adantl 482	. 2 |- ( ( J e. PNrm /\ A e. ( Clsd ` J ) ) -> ( A e. ran ( f e. ( J ^m NN ) |-> |^| ran f ) <-> E. f e. ( J ^m NN ) A = |^| ran f ) )
7	3, 6	mpbid 222	1 |- ( ( J e. PNrm /\ A e. ( Clsd ` J ) ) -> E. f e. ( J ^m NN ) A = |^| ran f )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   |^|cint 4475   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   NNcn 11020   Clsdccld 20820   Nrmcnrm 21114  PNrmcpnrm 21116

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-ov 6653  df-pnrm 21123

This theorem is referenced by:  pnrmopn  21147