Theorem crefdf 29915

Description: A formulation of crefi 29914 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Hypotheses

Ref	Expression
crefi.x	|- X = U. J
crefdf.b	|- B = CovHasRef A
crefdf.p	|- ( z e. A -> ph )

Assertion

Ref	Expression
crefdf	|- ( ( J e. B /\ C C_ J /\ X = U. C ) -> E. z e. ~P J ( ph /\ z Ref C ) )

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

Allowed substitution hints:   ph( z)   B( z)   X( z)

Proof of Theorem crefdf

Step	Hyp	Ref	Expression
1		crefdf.b	. . . 4 |- B = CovHasRef A
2	1	eleq2i 2693	. . 3 |- ( J e. B <-> J e. CovHasRef A )
3		crefi.x	. . . 4 |- X = U. J
4	3	crefi 29914	. . 3 |- ( ( J e. CovHasRef A /\ C C_ J /\ X = U. C ) -> E. z e. ( ~P J i^i A ) z Ref C )
5	2, 4	syl3an1b 1362	. 2 |- ( ( J e. B /\ C C_ J /\ X = U. C ) -> E. z e. ( ~P J i^i A ) z Ref C )
6		elin 3796	. . . . . 6 |- ( z e. ( ~P J i^i A ) <-> ( z e. ~P J /\ z e. A ) )
7		crefdf.p	. . . . . . 7 |- ( z e. A -> ph )
8	7	anim2i 593	. . . . . 6 |- ( ( z e. ~P J /\ z e. A ) -> ( z e. ~P J /\ ph ) )
9	6, 8	sylbi 207	. . . . 5 |- ( z e. ( ~P J i^i A ) -> ( z e. ~P J /\ ph ) )
10	9	anim1i 592	. . . 4 |- ( ( z e. ( ~P J i^i A ) /\ z Ref C ) -> ( ( z e. ~P J /\ ph ) /\ z Ref C ) )
11		anass 681	. . . 4 |- ( ( ( z e. ~P J /\ ph ) /\ z Ref C ) <-> ( z e. ~P J /\ ( ph /\ z Ref C ) ) )
12	10, 11	sylib 208	. . 3 |- ( ( z e. ( ~P J i^i A ) /\ z Ref C ) -> ( z e. ~P J /\ ( ph /\ z Ref C ) ) )
13	12	reximi2 3010	. 2 |- ( E. z e. ( ~P J i^i A ) z Ref C -> E. z e. ~P J ( ph /\ z Ref C ) )
14	5, 13	syl 17	1 |- ( ( J e. B /\ C C_ J /\ X = U. C ) -> E. z e. ~P J ( ph /\ z Ref C ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   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:  cmpfiref  29918  ldlfcntref  29921