Theorem iscref 29911

Description: The property that every open cover has an A refinement for the topological space J. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Hypothesis

Ref	Expression
iscref.x	|- X = U. J

Assertion

Ref	Expression
iscref	|- ( J e. CovHasRef A <-> ( J e. Top /\ A. y e. ~P J ( X = U. y -> E. z e. ( ~P J i^i A ) z Ref y ) ) )

Distinct variable groups:   y, A, z   y, J, z

Allowed substitution hints:   X( y, z)

Proof of Theorem iscref

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4161	. . 3 |- ( j = J -> ~P j = ~P J )
2		unieq 4444	. . . . . 6 |- ( j = J -> U. j = U. J )
3		iscref.x	. . . . . 6 |- X = U. J
4	2, 3	syl6eqr 2674	. . . . 5 |- ( j = J -> U. j = X )
5	4	eqeq1d 2624	. . . 4 |- ( j = J -> ( U. j = U. y <-> X = U. y ) )
6	1	ineq1d 3813	. . . . 5 |- ( j = J -> ( ~P j i^i A ) = ( ~P J i^i A ) )
7	6	rexeqdv 3145	. . . 4 |- ( j = J -> ( E. z e. ( ~P j i^i A ) z Ref y <-> E. z e. ( ~P J i^i A ) z Ref y ) )
8	5, 7	imbi12d 334	. . 3 |- ( j = J -> ( ( U. j = U. y -> E. z e. ( ~P j i^i A ) z Ref y ) <-> ( X = U. y -> E. z e. ( ~P J i^i A ) z Ref y ) ) )
9	1, 8	raleqbidv 3152	. 2 |- ( j = J -> ( A. y e. ~P j ( U. j = U. y -> E. z e. ( ~P j i^i A ) z Ref y ) <-> A. y e. ~P J ( X = U. y -> E. z e. ( ~P J i^i A ) z Ref y ) ) )
10		df-cref 29910	. 2 |- CovHasRef A = { j e. Top | A. y e. ~P j ( U. j = U. y -> E. z e. ( ~P j i^i A ) z Ref y ) }
11	9, 10	elrab2 3366	1 |- ( J e. CovHasRef A <-> ( J e. Top /\ A. y e. ~P J ( X = U. y -> E. z e. ( ~P J i^i A ) z Ref y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   Topctop 20698   Refcref 21305  CovHasRefccref 29909

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-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-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-cref 29910

This theorem is referenced by:  creftop  29913  crefi  29914  crefss  29916  cmpcref  29917  cmppcmp  29925  dispcmp  29926  pcmplfin  29927