Theorem bnj1415 33191

Description: Technical lemma for bnj60 33215. 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
bnj1415.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1415.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1415.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1415.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1415.5	|- D = { x e. A | -. E. f ta }
bnj1415.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1415.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1415.8	|- ( ta' <-> [. y / x ]. ta )
bnj1415.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1415.10	|- P = U. H

Assertion

Ref	Expression
bnj1415	|- ( ch -> dom P = trCl ( x , A , R ) )

Distinct variable groups:   A, f, x, y   B, f   y, C   y, D   R, f, x, y   f, d, x   ps, y   ta, y

Allowed substitution hints:   ps( x, f, d)   ch( x, y, f, d)   ta( x, f, d)   A( d)   B( x, y, d)   C( x, f, d)   D( x, f, d)   P( x, y, f, d)   R( d)   G( x, y, f, d)   H( x, y, f, d)   Y( x, y, f, d)   ta'( x, y, f, d)

Proof of Theorem bnj1415

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1415.7	. . . 4 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
2		bnj1415.6	. . . . 5 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
3	2	simplbi 460	. . . 4 |- ( ps -> R FrSe A )
4	1, 3	bnj835 32914	. . 3 |- ( ch -> R FrSe A )
5		bnj1415.5	. . . 4 |- D = { x e. A | -. E. f ta }
6	5, 1	bnj1212 32955	. . 3 |- ( ch -> x e. A )
7		eqid 2467	. . . 4 |- ( pred ( x , A , R ) u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) ) = ( pred ( x , A , R ) u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) )
8	7	bnj1414 33190	. . 3 |- ( ( R FrSe A /\ x e. A ) -> trCl ( x , A , R ) = ( pred ( x , A , R ) u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) ) )
9	4, 6, 8	syl2anc 661	. 2 |- ( ch -> trCl ( x , A , R ) = ( pred ( x , A , R ) u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) ) )
10		iunun 4406	. . . 4 |- U_ y e. pred ( x , A , R ) ( { y } u. trCl ( y , A , R ) ) = ( U_ y e. pred ( x , A , R ) { y } u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) )
11		iunid 4380	. . . . 5 |- U_ y e. pred ( x , A , R ) { y } = pred ( x , A , R )
12	11	uneq1i 3654	. . . 4 |- ( U_ y e. pred ( x , A , R ) { y } u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) ) = ( pred ( x , A , R ) u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) )
13	10, 12	eqtri 2496	. . 3 |- U_ y e. pred ( x , A , R ) ( { y } u. trCl ( y , A , R ) ) = ( pred ( x , A , R ) u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) )
14		bnj1415.1	. . . 4 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
15		bnj1415.2	. . . 4 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
16		bnj1415.3	. . . 4 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
17		bnj1415.4	. . . 4 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
18		bnj1415.8	. . . 4 |- ( ta' <-> [. y / x ]. ta )
19		bnj1415.9	. . . 4 |- H = { f | E. y e. pred ( x , A , R ) ta' }
20		bnj1415.10	. . . 4 |- P = U. H
21		biid 236	. . . 4 |- ( ( ch /\ z e. U_ y e. pred ( x , A , R ) ( { y } u. trCl ( y , A , R ) ) ) <-> ( ch /\ z e. U_ y e. pred ( x , A , R ) ( { y } u. trCl ( y , A , R ) ) ) )
22		biid 236	. . . 4 |- ( ( ( ch /\ z e. U_ y e. pred ( x , A , R ) ( { y } u. trCl ( y , A , R ) ) ) /\ y e. pred ( x , A , R ) /\ z e. ( { y } u. trCl ( y , A , R ) ) ) <-> ( ( ch /\ z e. U_ y e. pred ( x , A , R ) ( { y } u. trCl ( y , A , R ) ) ) /\ y e. pred ( x , A , R ) /\ z e. ( { y } u. trCl ( y , A , R ) ) ) )
23	14, 15, 16, 17, 5, 2, 1, 18, 19, 20, 21, 22	bnj1398 33187	. . 3 |- ( ch -> U_ y e. pred ( x , A , R ) ( { y } u. trCl ( y , A , R ) ) = dom P )
24	13, 23	syl5eqr 2522	. 2 |- ( ch -> ( pred ( x , A , R ) u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) ) = dom P )
25	9, 24	eqtr2d 2509	1 |- ( ch -> dom P = trCl ( x , A , R ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   {cab 2452   =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   [.wsbc 3331   u. cun 3474   C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   U_ciun 4325   class class class wbr 4447   dom cdm 4999   |` cres 5001   Fn wfn 5583   ` cfv 5588   predc-bnj14 32838   FrSe w-bnj15 32842   trClc-bnj18 32844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-reg 8018  ax-inf2 8058

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-bnj17 32837  df-bnj14 32839  df-bnj13 32841  df-bnj15 32843  df-bnj18 32845  df-bnj19 32847

This theorem is referenced by:  bnj1312  33211