Theorem crefi 29914

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
crefi.x	|- X = U. J

Assertion

Ref	Expression
crefi	|- ( ( J e. CovHasRef A /\ C C_ J /\ X = U. C ) -> E. z e. ( ~P J i^i A ) z Ref C )

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

Allowed substitution hint:   X( z)

Proof of Theorem crefi

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( J e. CovHasRef A /\ C C_ J /\ X = U. C ) -> J e. CovHasRef A )
2		simp2 1062	. . 3 |- ( ( J e. CovHasRef A /\ C C_ J /\ X = U. C ) -> C C_ J )
3	1, 2	sselpwd 4807	. 2 |- ( ( J e. CovHasRef A /\ C C_ J /\ X = U. C ) -> C e. ~P J )
4		crefi.x	. . . . 5 |- X = U. J
5	4	iscref 29911	. . . 4 |- ( 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 ) ) )
6	5	simprbi 480	. . 3 |- ( J e. CovHasRef A -> A. y e. ~P J ( X = U. y -> E. z e. ( ~P J i^i A ) z Ref y ) )
7	6	3ad2ant1 1082	. 2 |- ( ( J e. CovHasRef A /\ C C_ J /\ X = U. C ) -> A. y e. ~P J ( X = U. y -> E. z e. ( ~P J i^i A ) z Ref y ) )
8		simp3 1063	. 2 |- ( ( J e. CovHasRef A /\ C C_ J /\ X = U. C ) -> X = U. C )
9		unieq 4444	. . . . 5 |- ( y = C -> U. y = U. C )
10	9	eqeq2d 2632	. . . 4 |- ( y = C -> ( X = U. y <-> X = U. C ) )
11		breq2 4657	. . . . 5 |- ( y = C -> ( z Ref y <-> z Ref C ) )
12	11	rexbidv 3052	. . . 4 |- ( y = C -> ( E. z e. ( ~P J i^i A ) z Ref y <-> E. z e. ( ~P J i^i A ) z Ref C ) )
13	10, 12	imbi12d 334	. . 3 |- ( y = C -> ( ( X = U. y -> E. z e. ( ~P J i^i A ) z Ref y ) <-> ( X = U. C -> E. z e. ( ~P J i^i A ) z Ref C ) ) )
14	13	rspcv 3305	. 2 |- ( C e. ~P J -> ( A. y e. ~P J ( X = U. y -> E. z e. ( ~P J i^i A ) z Ref y ) -> ( X = U. C -> E. z e. ( ~P J i^i A ) z Ref C ) ) )
15	3, 7, 8, 14	syl3c 66	1 |- ( ( J e. CovHasRef A /\ C C_ J /\ X = U. C ) -> E. z e. ( ~P J i^i A ) z Ref C )

Syntax hints:   -> wi 4   /\ w3a 1037   = 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  ax-sep 4781

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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-cref 29910

This theorem is referenced by:  crefdf  29915