Theorem crefeq 29912

Description: Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
crefeq	|- ( A = B -> CovHasRef A = CovHasRef B )

Proof of Theorem crefeq

Dummy variables j y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq2 3808	. . . . . 6 |- ( A = B -> ( ~P j i^i A ) = ( ~P j i^i B ) )
2	1	rexeqdv 3145	. . . . 5 |- ( A = B -> ( E. z e. ( ~P j i^i A ) z Ref y <-> E. z e. ( ~P j i^i B ) z Ref y ) )
3	2	imbi2d 330	. . . 4 |- ( A = B -> ( ( U. j = U. y -> E. z e. ( ~P j i^i A ) z Ref y ) <-> ( U. j = U. y -> E. z e. ( ~P j i^i B ) z Ref y ) ) )
4	3	ralbidv 2986	. . 3 |- ( A = B -> ( 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 ( U. j = U. y -> E. z e. ( ~P j i^i B ) z Ref y ) ) )
5	4	rabbidv 3189	. 2 |- ( A = B -> { j e. Top | A. y e. ~P j ( U. j = U. y -> E. z e. ( ~P j i^i A ) z Ref y ) } = { j e. Top | A. y e. ~P j ( U. j = U. y -> E. z e. ( ~P j i^i B ) z Ref y ) } )
6		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 ) }
7		df-cref 29910	. 2 |- CovHasRef B = { j e. Top | A. y e. ~P j ( U. j = U. y -> E. z e. ( ~P j i^i B ) z Ref y ) }
8	5, 6, 7	3eqtr4g 2681	1 |- ( A = B -> CovHasRef A = CovHasRef B )

Syntax hints:   -> wi 4   = wceq 1483   A.wral 2912   E.wrex 2913   {crab 2916   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-cref 29910

This theorem is referenced by:  ispcmp  29924