Theorem bnj1445 31112

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
bnj1445.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1445.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1445.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1445.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1445.5	|- D = { x e. A | -. E. f ta }
bnj1445.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1445.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1445.8	|- ( ta' <-> [. y / x ]. ta )
bnj1445.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1445.10	|- P = U. H
bnj1445.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1445.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1445.13	|- W = <. z , ( Q |` pred ( z , A , R ) ) >.
bnj1445.14	|- E = ( { x } u. trCl ( x , A , R ) )
bnj1445.15	|- ( ch -> P Fn trCl ( x , A , R ) )
bnj1445.16	|- ( ch -> Q Fn ( { x } u. trCl ( x , A , R ) ) )
bnj1445.17	|- ( th <-> ( ch /\ z e. E ) )
bnj1445.18	|- ( et <-> ( th /\ z e. { x } ) )
bnj1445.19	|- ( ze <-> ( th /\ z e. trCl ( x , A , R ) ) )
bnj1445.20	|- ( rh <-> ( ze /\ f e. H /\ z e. dom f ) )
bnj1445.21	|- ( si <-> ( rh /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
bnj1445.22	|- ( ph <-> ( si /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
bnj1445.23	|- X = <. z , ( f |` pred ( z , A , R ) ) >.

Assertion

Ref	Expression
bnj1445	|- ( si -> A. d si )

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

Allowed substitution hints:   ph( x, y, z, f, d)   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)   si( x, y, z, f, d)   rh( x, y, z, f, d)   A( y, z, f)   B( x, y, z, 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( y, z, f)   E( x, y, z, f)   G( x, y, z, f, d)   H( x, y, z, f, d)   W( x, y, z, f, d)   X( 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 bnj1445

Step	Hyp	Ref	Expression
1		bnj1445.21	. 2 |- ( si <-> ( rh /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
2		bnj1445.20	. . . . 5 |- ( rh <-> ( ze /\ f e. H /\ z e. dom f ) )
3		bnj1445.19	. . . . . . 7 |- ( ze <-> ( th /\ z e. trCl ( x , A , R ) ) )
4		bnj1445.17	. . . . . . . . 9 |- ( th <-> ( ch /\ z e. E ) )
5		bnj1445.7	. . . . . . . . . . . . 13 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
6		bnj1445.6	. . . . . . . . . . . . . . 15 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
7		nfv 1843	. . . . . . . . . . . . . . . 16 |- F/ d R FrSe A
8		bnj1445.5	. . . . . . . . . . . . . . . . . 18 |- D = { x e. A | -. E. f ta }
9		bnj1445.4	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
10		bnj1445.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
11		nfre1 3005	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- F/ d E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
12	11	nfab 2769	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- F/_ d { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
13	10, 12	nfcxfr 2762	. . . . . . . . . . . . . . . . . . . . . . . 24 |- F/_ d C
14	13	nfcri 2758	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ d f e. C
15		nfv 1843	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ d dom f = ( { x } u. trCl ( x , A , R ) )
16	14, 15	nfan 1828	. . . . . . . . . . . . . . . . . . . . . 22 |- F/ d ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) )
17	9, 16	nfxfr 1779	. . . . . . . . . . . . . . . . . . . . 21 |- F/ d ta
18	17	nfex 2154	. . . . . . . . . . . . . . . . . . . 20 |- F/ d E. f ta
19	18	nfn 1784	. . . . . . . . . . . . . . . . . . 19 |- F/ d -. E. f ta
20		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 |- F/_ d A
21	19, 20	nfrab 3123	. . . . . . . . . . . . . . . . . 18 |- F/_ d { x e. A | -. E. f ta }
22	8, 21	nfcxfr 2762	. . . . . . . . . . . . . . . . 17 |- F/_ d D
23		nfcv 2764	. . . . . . . . . . . . . . . . 17 |- F/_ d (/)
24	22, 23	nfne 2894	. . . . . . . . . . . . . . . 16 |- F/ d D =/= (/)
25	7, 24	nfan 1828	. . . . . . . . . . . . . . 15 |- F/ d ( R FrSe A /\ D =/= (/) )
26	6, 25	nfxfr 1779	. . . . . . . . . . . . . 14 |- F/ d ps
27	22	nfcri 2758	. . . . . . . . . . . . . 14 |- F/ d x e. D
28		nfv 1843	. . . . . . . . . . . . . . 15 |- F/ d -. y R x
29	22, 28	nfral 2945	. . . . . . . . . . . . . 14 |- F/ d A. y e. D -. y R x
30	26, 27, 29	nf3an 1831	. . . . . . . . . . . . 13 |- F/ d ( ps /\ x e. D /\ A. y e. D -. y R x )
31	5, 30	nfxfr 1779	. . . . . . . . . . . 12 |- F/ d ch
32	31	nf5ri 2065	. . . . . . . . . . 11 |- ( ch -> A. d ch )
33	32	bnj1351 30897	. . . . . . . . . 10 |- ( ( ch /\ z e. E ) -> A. d ( ch /\ z e. E ) )
34	33	nf5i 2024	. . . . . . . . 9 |- F/ d ( ch /\ z e. E )
35	4, 34	nfxfr 1779	. . . . . . . 8 |- F/ d th
36		nfv 1843	. . . . . . . 8 |- F/ d z e. trCl ( x , A , R )
37	35, 36	nfan 1828	. . . . . . 7 |- F/ d ( th /\ z e. trCl ( x , A , R ) )
38	3, 37	nfxfr 1779	. . . . . 6 |- F/ d ze
39		bnj1445.9	. . . . . . . 8 |- H = { f | E. y e. pred ( x , A , R ) ta' }
40		nfcv 2764	. . . . . . . . . 10 |- F/_ d pred ( x , A , R )
41		bnj1445.8	. . . . . . . . . . 11 |- ( ta' <-> [. y / x ]. ta )
42		nfcv 2764	. . . . . . . . . . . 12 |- F/_ d y
43	42, 17	nfsbc 3457	. . . . . . . . . . 11 |- F/ d [. y / x ]. ta
44	41, 43	nfxfr 1779	. . . . . . . . . 10 |- F/ d ta'
45	40, 44	nfrex 3007	. . . . . . . . 9 |- F/ d E. y e. pred ( x , A , R ) ta'
46	45	nfab 2769	. . . . . . . 8 |- F/_ d { f | E. y e. pred ( x , A , R ) ta' }
47	39, 46	nfcxfr 2762	. . . . . . 7 |- F/_ d H
48	47	nfcri 2758	. . . . . 6 |- F/ d f e. H
49		nfv 1843	. . . . . 6 |- F/ d z e. dom f
50	38, 48, 49	nf3an 1831	. . . . 5 |- F/ d ( ze /\ f e. H /\ z e. dom f )
51	2, 50	nfxfr 1779	. . . 4 |- F/ d rh
52	51	nf5ri 2065	. . 3 |- ( rh -> A. d rh )
53		ax-5 1839	. . 3 |- ( y e. pred ( x , A , R ) -> A. d y e. pred ( x , A , R ) )
54	14	nf5ri 2065	. . 3 |- ( f e. C -> A. d f e. C )
55		ax-5 1839	. . 3 |- ( dom f = ( { y } u. trCl ( y , A , R ) ) -> A. d dom f = ( { y } u. trCl ( y , A , R ) ) )
56	52, 53, 54, 55	bnj982 30849	. 2 |- ( ( rh /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) -> A. d ( rh /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
57	1, 56	hbxfrbi 1752	1 |- ( si -> A. d si )

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   /\ w-bnj17 30752   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-rex 2918  df-rab 2921  df-sbc 3436  df-bnj17 30753

This theorem is referenced by:  bnj1450  31118