Theorem creftop 29913

Description: A space where every open cover has an A refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
creftop	|- ( J e. CovHasRef A -> J e. Top )

Proof of Theorem creftop

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- U. J = U. J
2	1	iscref 29911	. 2 |- ( J e. 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 ) ) )
3	2	simplbi 476	1 |- ( J e. CovHasRef A -> J e. Top )

Syntax hints:   -> wi 4   = 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: (None)