Theorem ispcmp 29924

Description: The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
ispcmp	|- ( J e. Paracomp <-> J e. CovHasRef ( LocFin ` J ) )

Proof of Theorem ispcmp

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( J e. Paracomp -> J e. _V )
2		elex 3212	. 2 |- ( J e. CovHasRef ( LocFin ` J ) -> J e. _V )
3		id 22	. . . 4 |- ( j = J -> j = J )
4		fveq2 6191	. . . . 5 |- ( j = J -> ( LocFin ` j ) = ( LocFin ` J ) )
5		crefeq 29912	. . . . 5 |- ( ( LocFin ` j ) = ( LocFin ` J ) -> CovHasRef ( LocFin ` j ) = CovHasRef ( LocFin ` J ) )
6	4, 5	syl 17	. . . 4 |- ( j = J -> CovHasRef ( LocFin ` j ) = CovHasRef ( LocFin ` J ) )
7	3, 6	eleq12d 2695	. . 3 |- ( j = J -> ( j e. CovHasRef ( LocFin ` j ) <-> J e. CovHasRef ( LocFin ` J ) ) )
8		df-pcmp 29923	. . 3 |- Paracomp = { j | j e. CovHasRef ( LocFin ` j ) }
9	7, 8	elab2g 3353	. 2 |- ( J e. _V -> ( J e. Paracomp <-> J e. CovHasRef ( LocFin ` J ) ) )
10	1, 2, 9	pm5.21nii 368	1 |- ( J e. Paracomp <-> J e. CovHasRef ( LocFin ` J ) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888   LocFinclocfin 21307  CovHasRefccref 29909  Paracompcpcmp 29922

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

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-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-iota 5851  df-fv 5896  df-cref 29910  df-pcmp 29923

This theorem is referenced by:  cmppcmp  29925  dispcmp  29926  pcmplfin  29927