Theorem bnj1449 31116

Description: Technical lemma for bnj60 31130. 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
bnj1449.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1449.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1449.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1449.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1449.5	|- D = { x e. A | -. E. f ta }
bnj1449.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1449.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1449.8	|- ( ta' <-> [. y / x ]. ta )
bnj1449.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1449.10	|- P = U. H
bnj1449.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1449.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1449.13	|- W = <. z , ( Q |` pred ( z , A , R ) ) >.
bnj1449.14	|- E = ( { x } u. trCl ( x , A , R ) )
bnj1449.15	|- ( ch -> P Fn trCl ( x , A , R ) )
bnj1449.16	|- ( ch -> Q Fn ( { x } u. trCl ( x , A , R ) ) )
bnj1449.17	|- ( th <-> ( ch /\ z e. E ) )
bnj1449.18	|- ( et <-> ( th /\ z e. { x } ) )
bnj1449.19	|- ( ze <-> ( th /\ z e. trCl ( x , A , R ) ) )

Assertion

Ref	Expression
bnj1449	|- ( ze -> A. f ze )

Distinct variable groups:   A, f   f, E   R, f   x, f   y, f   z, f

Allowed substitution hints:   ps( x, y, z, f, d)   ch( x, y, z, f, d)   th( x, y, z, f, d)   ta( x, y, z, f, d)   et( x, y, z, f, d)   ze( x, y, z, f, d)   A( x, y, z, d)   B( x, y, z, f, d)   C( x, y, z, f, d)   D( x, y, z, f, d)   P( x, y, z, f, d)   Q( x, y, z, f, d)   R( x, y, z, d)   E( x, y, z, d)   G( x, y, z, f, d)   H( x, y, z, f, d)   W( x, y, z, f, d)   Y( x, y, z, f, d)   Z( x, y, z, f, d)   ta'( x, y, z, f, d)

Proof of Theorem bnj1449

Step	Hyp	Ref	Expression
1		bnj1449.19	. . 3 |- ( ze <-> ( th /\ z e. trCl ( x , A , R ) ) )
2		bnj1449.17	. . . . 5 |- ( th <-> ( ch /\ z e. E ) )
3		bnj1449.7	. . . . . . 7 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
4		bnj1449.6	. . . . . . . . 9 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
5		nfv 1843	. . . . . . . . . 10 |- F/ f R FrSe A
6		bnj1449.5	. . . . . . . . . . . 12 |- D = { x e. A | -. E. f ta }
7		nfe1 2027	. . . . . . . . . . . . . 14 |- F/ f E. f ta
8	7	nfn 1784	. . . . . . . . . . . . 13 |- F/ f -. E. f ta
9		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ f A
10	8, 9	nfrab 3123	. . . . . . . . . . . 12 |- F/_ f { x e. A | -. E. f ta }
11	6, 10	nfcxfr 2762	. . . . . . . . . . 11 |- F/_ f D
12		nfcv 2764	. . . . . . . . . . 11 |- F/_ f (/)
13	11, 12	nfne 2894	. . . . . . . . . 10 |- F/ f D =/= (/)
14	5, 13	nfan 1828	. . . . . . . . 9 |- F/ f ( R FrSe A /\ D =/= (/) )
15	4, 14	nfxfr 1779	. . . . . . . 8 |- F/ f ps
16	11	nfcri 2758	. . . . . . . 8 |- F/ f x e. D
17		nfv 1843	. . . . . . . . 9 |- F/ f -. y R x
18	11, 17	nfral 2945	. . . . . . . 8 |- F/ f A. y e. D -. y R x
19	15, 16, 18	nf3an 1831	. . . . . . 7 |- F/ f ( ps /\ x e. D /\ A. y e. D -. y R x )
20	3, 19	nfxfr 1779	. . . . . 6 |- F/ f ch
21		nfv 1843	. . . . . 6 |- F/ f z e. E
22	20, 21	nfan 1828	. . . . 5 |- F/ f ( ch /\ z e. E )
23	2, 22	nfxfr 1779	. . . 4 |- F/ f th
24		nfv 1843	. . . 4 |- F/ f z e. trCl ( x , A , R )
25	23, 24	nfan 1828	. . 3 |- F/ f ( th /\ z e. trCl ( x , A , R ) )
26	1, 25	nfxfr 1779	. 2 |- F/ f ze
27	26	nf5ri 2065	1 |- ( ze -> A. f ze )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   [.wsbc 3435   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

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-rab 2921

This theorem is referenced by:  bnj1450  31118