Theorem bnj1415 33962

Description: Technical lemma for bnj60 33986. 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 33685	. . 3 |- ( ch -> R FrSe A )
5		bnj1415.5	. . . 4 |- D = { x e. A | -. E. f ta }
6	5, 1	bnj1212 33726	. . 3 |- ( ch -> x e. A )
7		eqid 2443	. . . 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 33961	. . 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 4396	. . . 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 4370	. . . . 5 |- U_ y e. pred ( x , A , R ) { y } = pred ( x , A , R )
12	11	uneq1i 3639	. . . 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 2472	. . 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 33958	. . 3 |- ( ch -> U_ y e. pred ( x , A , R ) ( { y } u. trCl ( y , A , R ) ) = dom P )
24	13, 23	syl5eqr 2498	. 2 |- ( ch -> ( pred ( x , A , R ) u. U_ y e. pred ( x , A , R ) trCl ( y , A , R ) ) = dom P )
25	9, 24	eqtr2d 2485	1 |- ( ch -> dom P = trCl ( x , A , R ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   E.wex 1599   e. wcel 1804   {cab 2428   =/= wne 2638   A.wral 2793   E.wrex 2794   {crab 2797   [.wsbc 3313   u. cun 3459   C_ wss 3461   (/)c0 3770   {csn 4014   <.cop 4020   U.cuni 4234   U_ciun 4315   class class class wbr 4437   dom cdm 4989   |` cres 4991   Fn wfn 5573   ` cfv 5578   predc-bnj14 33608   FrSe w-bnj15 33612   trClc-bnj18 33614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-reg 8021  ax-inf2 8061

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-1o 7132  df-bnj17 33607  df-bnj14 33609  df-bnj13 33611  df-bnj15 33613  df-bnj18 33615  df-bnj19 33617

This theorem is referenced by:  bnj1312  33982