Theorem crefss 29916

Description: The "every open cover has an A refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
crefss	|- ( A C_ B -> CovHasRef A C_ CovHasRef B )

Proof of Theorem crefss

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

Step	Hyp	Ref	Expression
1		sslin 3839	. . . . . . 7 |- ( A C_ B -> ( ~P j i^i A ) C_ ( ~P j i^i B ) )
2		ssrexv 3667	. . . . . . 7 |- ( ( ~P j i^i A ) C_ ( ~P j i^i B ) -> ( E. z e. ( ~P j i^i A ) z Ref y -> E. z e. ( ~P j i^i B ) z Ref y ) )
3	1, 2	syl 17	. . . . . 6 |- ( A C_ B -> ( E. z e. ( ~P j i^i A ) z Ref y -> E. z e. ( ~P j i^i B ) z Ref y ) )
4	3	imim2d 57	. . . . 5 |- ( A C_ 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 ) ) )
5	4	ralimdv 2963	. . . 4 |- ( A C_ 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 ) ) )
6	5	anim2d 589	. . 3 |- ( A C_ 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 ) ) ) )
7		eqid 2622	. . . 4 |- U. j = U. j
8	7	iscref 29911	. . 3 |- ( 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 ) ) )
9	7	iscref 29911	. . 3 |- ( j e. 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 ) ) )
10	6, 8, 9	3imtr4g 285	. 2 |- ( A C_ B -> ( j e. CovHasRef A -> j e. CovHasRef B ) )
11	10	ssrdv 3609	1 |- ( A C_ B -> CovHasRef A C_ CovHasRef B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~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)