Theorem bnj1518 31132

Description: Technical lemma for bnj1500 31136. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1518.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1518.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1518.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1518.4	|- F = U. C
bnj1518.5	|- ( ph <-> ( R FrSe A /\ x e. A ) )
bnj1518.6	|- ( ps <-> ( ph /\ f e. C /\ x e. dom f ) )

Assertion

Ref	Expression
bnj1518	|- ( ps -> A. d ps )

Distinct variable groups:   f, d   ph, d   x, d

Allowed substitution hints:   ph( x, f)   ps( x, f, d)   A( x, f, d)   B( x, f, d)   C( x, f, d)   R( x, f, d)   F( x, f, d)   G( x, f, d)   Y( x, f, d)

Proof of Theorem bnj1518

Step	Hyp	Ref	Expression
1		bnj1518.6	. . 3 |- ( ps <-> ( ph /\ f e. C /\ x e. dom f ) )
2		nfv 1843	. . . 4 |- F/ d ph
3		bnj1518.3	. . . . . 6 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		nfre1 3005	. . . . . . 7 |- F/ d E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
5	4	nfab 2769	. . . . . 6 |- F/_ d { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
6	3, 5	nfcxfr 2762	. . . . 5 |- F/_ d C
7	6	nfcri 2758	. . . 4 |- F/ d f e. C
8		nfv 1843	. . . 4 |- F/ d x e. dom f
9	2, 7, 8	nf3an 1831	. . 3 |- F/ d ( ph /\ f e. C /\ x e. dom f )
10	1, 9	nfxfr 1779	. 2 |- F/ d ps
11	10	nf5ri 2065	1 |- ( ps -> A. d ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   <.cop 4183   U.cuni 4436   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758

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-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-rex 2918

This theorem is referenced by:  bnj1501  31135