Theorem bnj1446 31113

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
bnj1446.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1446.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1446.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1446.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1446.5	|- D = { x e. A | -. E. f ta }
bnj1446.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1446.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1446.8	|- ( ta' <-> [. y / x ]. ta )
bnj1446.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1446.10	|- P = U. H
bnj1446.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1446.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1446.13	|- W = <. z , ( Q |` pred ( z , A , R ) ) >.

Assertion

Ref	Expression
bnj1446	|- ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )

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

Allowed substitution hints:   ps( x, y, z, f, d)   ch( x, y, z, f, d)   ta( 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)   G( x, y, z, f)   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 bnj1446

Step	Hyp	Ref	Expression
1		bnj1446.12	. . . . 5 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
2		bnj1446.10	. . . . . . 7 |- P = U. H
3		bnj1446.9	. . . . . . . . 9 |- H = { f | E. y e. pred ( x , A , R ) ta' }
4		nfcv 2764	. . . . . . . . . . 11 |- F/_ d pred ( x , A , R )
5		bnj1446.8	. . . . . . . . . . . 12 |- ( ta' <-> [. y / x ]. ta )
6		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ d y
7		bnj1446.4	. . . . . . . . . . . . . 14 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
8		bnj1446.3	. . . . . . . . . . . . . . . . 17 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
9		nfre1 3005	. . . . . . . . . . . . . . . . . 18 |- F/ d E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
10	9	nfab 2769	. . . . . . . . . . . . . . . . 17 |- F/_ d { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
11	8, 10	nfcxfr 2762	. . . . . . . . . . . . . . . 16 |- F/_ d C
12	11	nfcri 2758	. . . . . . . . . . . . . . 15 |- F/ d f e. C
13		nfv 1843	. . . . . . . . . . . . . . 15 |- F/ d dom f = ( { x } u. trCl ( x , A , R ) )
14	12, 13	nfan 1828	. . . . . . . . . . . . . 14 |- F/ d ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) )
15	7, 14	nfxfr 1779	. . . . . . . . . . . . 13 |- F/ d ta
16	6, 15	nfsbc 3457	. . . . . . . . . . . 12 |- F/ d [. y / x ]. ta
17	5, 16	nfxfr 1779	. . . . . . . . . . 11 |- F/ d ta'
18	4, 17	nfrex 3007	. . . . . . . . . 10 |- F/ d E. y e. pred ( x , A , R ) ta'
19	18	nfab 2769	. . . . . . . . 9 |- F/_ d { f | E. y e. pred ( x , A , R ) ta' }
20	3, 19	nfcxfr 2762	. . . . . . . 8 |- F/_ d H
21	20	nfuni 4442	. . . . . . 7 |- F/_ d U. H
22	2, 21	nfcxfr 2762	. . . . . 6 |- F/_ d P
23		nfcv 2764	. . . . . . . 8 |- F/_ d x
24		nfcv 2764	. . . . . . . . 9 |- F/_ d G
25		bnj1446.11	. . . . . . . . . 10 |- Z = <. x , ( P |` pred ( x , A , R ) ) >.
26	22, 4	nfres 5398	. . . . . . . . . . 11 |- F/_ d ( P |` pred ( x , A , R ) )
27	23, 26	nfop 4418	. . . . . . . . . 10 |- F/_ d <. x , ( P |` pred ( x , A , R ) ) >.
28	25, 27	nfcxfr 2762	. . . . . . . . 9 |- F/_ d Z
29	24, 28	nffv 6198	. . . . . . . 8 |- F/_ d ( G ` Z )
30	23, 29	nfop 4418	. . . . . . 7 |- F/_ d <. x , ( G ` Z ) >.
31	30	nfsn 4242	. . . . . 6 |- F/_ d { <. x , ( G ` Z ) >. }
32	22, 31	nfun 3769	. . . . 5 |- F/_ d ( P u. { <. x , ( G ` Z ) >. } )
33	1, 32	nfcxfr 2762	. . . 4 |- F/_ d Q
34		nfcv 2764	. . . 4 |- F/_ d z
35	33, 34	nffv 6198	. . 3 |- F/_ d ( Q ` z )
36		bnj1446.13	. . . . 5 |- W = <. z , ( Q |` pred ( z , A , R ) ) >.
37		nfcv 2764	. . . . . . 7 |- F/_ d pred ( z , A , R )
38	33, 37	nfres 5398	. . . . . 6 |- F/_ d ( Q |` pred ( z , A , R ) )
39	34, 38	nfop 4418	. . . . 5 |- F/_ d <. z , ( Q |` pred ( z , A , R ) ) >.
40	36, 39	nfcxfr 2762	. . . 4 |- F/_ d W
41	24, 40	nffv 6198	. . 3 |- F/_ d ( G ` W )
42	35, 41	nfeq 2776	. 2 |- F/ d ( Q ` z ) = ( G ` W )
43	42	nf5ri 2065	1 |- ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj1450  31118  bnj1463  31123