Theorem bnj1525 31137

Description: Technical lemma for bnj1522 31140. 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
bnj1525.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1525.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1525.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1525.4	|- F = U. C
bnj1525.5	|- ( ph <-> ( R FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) ) )
bnj1525.6	|- ( ps <-> ( ph /\ F =/= H ) )

Assertion

Ref	Expression
bnj1525	|- ( ps -> A. x ps )

Distinct variable groups:   x, A   x, H   x, R   x, d   x, f

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

Proof of Theorem bnj1525

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1525.6	. . 3 |- ( ps <-> ( ph /\ F =/= H ) )
2		bnj1525.5	. . . . 5 |- ( ph <-> ( R FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) ) )
3		nfv 1843	. . . . . 6 |- F/ x R FrSe A
4		nfv 1843	. . . . . 6 |- F/ x H Fn A
5		nfra1 2941	. . . . . 6 |- F/ x A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. )
6	3, 4, 5	nf3an 1831	. . . . 5 |- F/ x ( R FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) )
7	2, 6	nfxfr 1779	. . . 4 |- F/ x ph
8		bnj1525.4	. . . . . 6 |- F = U. C
9		bnj1525.3	. . . . . . . . 9 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
10		bnj1525.1	. . . . . . . . . 10 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
11	10	bnj1309 31090	. . . . . . . . 9 |- ( w e. B -> A. x w e. B )
12	9, 11	bnj1307 31091	. . . . . . . 8 |- ( w e. C -> A. x w e. C )
13	12	nfcii 2755	. . . . . . 7 |- F/_ x C
14	13	nfuni 4442	. . . . . 6 |- F/_ x U. C
15	8, 14	nfcxfr 2762	. . . . 5 |- F/_ x F
16		nfcv 2764	. . . . 5 |- F/_ x H
17	15, 16	nfne 2894	. . . 4 |- F/ x F =/= H
18	7, 17	nfan 1828	. . 3 |- F/ x ( ph /\ F =/= H )
19	1, 18	nfxfr 1779	. 2 |- F/ x ps
20	19	nf5ri 2065	1 |- ( ps -> A. x ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   <.cop 4183   U.cuni 4436   |` 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-ne 2795  df-ral 2917  df-rex 2918  df-uni 4437

This theorem is referenced by:  bnj1523  31139