Theorem bnj1423 31119

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
bnj1423.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1423.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1423.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1423.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1423.5	|- D = { x e. A | -. E. f ta }
bnj1423.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1423.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1423.8	|- ( ta' <-> [. y / x ]. ta )
bnj1423.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1423.10	|- P = U. H
bnj1423.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1423.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1423.13	|- W = <. z , ( Q |` pred ( z , A , R ) ) >.
bnj1423.14	|- E = ( { x } u. trCl ( x , A , R ) )
bnj1423.15	|- ( ch -> P Fn trCl ( x , A , R ) )
bnj1423.16	|- ( ch -> Q Fn ( { x } u. trCl ( x , A , R ) ) )

Assertion

Ref	Expression
bnj1423	|- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )

Distinct variable groups:   A, d, f, x, y, z   B, f   y, D   E, d, f, y   G, d, f, x, y, z   R, d, f, x, y, z   z, Y   ch, z   ps, y

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

Proof of Theorem bnj1423

Step	Hyp	Ref	Expression
1		bnj1423.1	. . . 4 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
2		bnj1423.2	. . . 4 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
3		bnj1423.3	. . . 4 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1423.4	. . . 4 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
5		bnj1423.5	. . . 4 |- D = { x e. A | -. E. f ta }
6		bnj1423.6	. . . 4 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
7		bnj1423.7	. . . 4 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8		bnj1423.8	. . . 4 |- ( ta' <-> [. y / x ]. ta )
9		bnj1423.9	. . . 4 |- H = { f | E. y e. pred ( x , A , R ) ta' }
10		bnj1423.10	. . . 4 |- P = U. H
11		bnj1423.11	. . . 4 |- Z = <. x , ( P |` pred ( x , A , R ) ) >.
12		bnj1423.12	. . . 4 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
13		bnj1423.13	. . . 4 |- W = <. z , ( Q |` pred ( z , A , R ) ) >.
14		bnj1423.14	. . . 4 |- E = ( { x } u. trCl ( x , A , R ) )
15		bnj1423.15	. . . 4 |- ( ch -> P Fn trCl ( x , A , R ) )
16		bnj1423.16	. . . 4 |- ( ch -> Q Fn ( { x } u. trCl ( x , A , R ) ) )
17		biid 251	. . . 4 |- ( ( ch /\ z e. E ) <-> ( ch /\ z e. E ) )
18		biid 251	. . . 4 |- ( ( ( ch /\ z e. E ) /\ z e. { x } ) <-> ( ( ch /\ z e. E ) /\ z e. { x } ) )
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	bnj1442 31117	. . 3 |- ( ( ( ch /\ z e. E ) /\ z e. { x } ) -> ( Q ` z ) = ( G ` W ) )
20		biid 251	. . . 4 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) <-> ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) )
21		biid 251	. . . 4 |- ( ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) <-> ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) )
22		biid 251	. . . 4 |- ( ( ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) <-> ( ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
23		biid 251	. . . 4 |- ( ( ( ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> ( ( ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
24		eqid 2622	. . . 4 |- <. z , ( f |` pred ( z , A , R ) ) >. = <. z , ( f |` pred ( z , A , R ) ) >.
25	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24	bnj1450 31118	. . 3 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) -> ( Q ` z ) = ( G ` W ) )
26	14	bnj1424 30909	. . . 4 |- ( z e. E -> ( z e. { x } \/ z e. trCl ( x , A , R ) ) )
27	26	adantl 482	. . 3 |- ( ( ch /\ z e. E ) -> ( z e. { x } \/ z e. trCl ( x , A , R ) ) )
28	19, 25, 27	mpjaodan 827	. 2 |- ( ( ch /\ z e. E ) -> ( Q ` z ) = ( G ` W ) )
29	28	ralrimiva 2966	1 |- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = 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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761

This theorem is referenced by:  bnj1312  31126