Theorem bnj1452 31120

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
bnj1452.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1452.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1452.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1452.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1452.5	|- D = { x e. A | -. E. f ta }
bnj1452.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1452.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1452.8	|- ( ta' <-> [. y / x ]. ta )
bnj1452.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1452.10	|- P = U. H
bnj1452.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1452.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1452.13	|- W = <. z , ( Q |` pred ( z , A , R ) ) >.
bnj1452.14	|- E = ( { x } u. trCl ( x , A , R ) )

Assertion

Ref	Expression
bnj1452	|- ( ch -> E e. B )

Distinct variable groups:   A, d, x, z   E, d, z   R, d, x, z   ch, z

Allowed substitution hints:   ps( x, y, z, f, d)   ch( x, y, f, d)   ta( x, y, z, f, d)   A( y, f)   B( x, y, z, f, 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, f)   E( x, y, f)   G( x, y, z, f, d)   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 bnj1452

Step	Hyp	Ref	Expression
1		bnj1452.14	. . 3 |- E = ( { x } u. trCl ( x , A , R ) )
2		bnj1452.5	. . . . . 6 |- D = { x e. A | -. E. f ta }
3		bnj1452.7	. . . . . 6 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
4	2, 3	bnj1212 30870	. . . . 5 |- ( ch -> x e. A )
5	4	snssd 4340	. . . 4 |- ( ch -> { x } C_ A )
6		bnj1147 31062	. . . . 5 |- trCl ( x , A , R ) C_ A
7	6	a1i 11	. . . 4 |- ( ch -> trCl ( x , A , R ) C_ A )
8	5, 7	unssd 3789	. . 3 |- ( ch -> ( { x } u. trCl ( x , A , R ) ) C_ A )
9	1, 8	syl5eqss 3649	. 2 |- ( ch -> E C_ A )
10		elsni 4194	. . . . . . . 8 |- ( z e. { x } -> z = x )
11	10	adantl 482	. . . . . . 7 |- ( ( ( ch /\ z e. E ) /\ z e. { x } ) -> z = x )
12		bnj602 30985	. . . . . . 7 |- ( z = x -> pred ( z , A , R ) = pred ( x , A , R ) )
13	11, 12	syl 17	. . . . . 6 |- ( ( ( ch /\ z e. E ) /\ z e. { x } ) -> pred ( z , A , R ) = pred ( x , A , R ) )
14		bnj1452.6	. . . . . . . . . 10 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
15	14	simplbi 476	. . . . . . . . 9 |- ( ps -> R FrSe A )
16	3, 15	bnj835 30829	. . . . . . . 8 |- ( ch -> R FrSe A )
17		bnj906 31000	. . . . . . . 8 |- ( ( R FrSe A /\ x e. A ) -> pred ( x , A , R ) C_ trCl ( x , A , R ) )
18	16, 4, 17	syl2anc 693	. . . . . . 7 |- ( ch -> pred ( x , A , R ) C_ trCl ( x , A , R ) )
19	18	ad2antrr 762	. . . . . 6 |- ( ( ( ch /\ z e. E ) /\ z e. { x } ) -> pred ( x , A , R ) C_ trCl ( x , A , R ) )
20	13, 19	eqsstrd 3639	. . . . 5 |- ( ( ( ch /\ z e. E ) /\ z e. { x } ) -> pred ( z , A , R ) C_ trCl ( x , A , R ) )
21		ssun4 3779	. . . . . 6 |- ( pred ( z , A , R ) C_ trCl ( x , A , R ) -> pred ( z , A , R ) C_ ( { x } u. trCl ( x , A , R ) ) )
22	21, 1	syl6sseqr 3652	. . . . 5 |- ( pred ( z , A , R ) C_ trCl ( x , A , R ) -> pred ( z , A , R ) C_ E )
23	20, 22	syl 17	. . . 4 |- ( ( ( ch /\ z e. E ) /\ z e. { x } ) -> pred ( z , A , R ) C_ E )
24	16	ad2antrr 762	. . . . . . 7 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) -> R FrSe A )
25		simpr 477	. . . . . . . 8 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) -> z e. trCl ( x , A , R ) )
26	6, 25	bnj1213 30869	. . . . . . 7 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) -> z e. A )
27		bnj906 31000	. . . . . . 7 |- ( ( R FrSe A /\ z e. A ) -> pred ( z , A , R ) C_ trCl ( z , A , R ) )
28	24, 26, 27	syl2anc 693	. . . . . 6 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) -> pred ( z , A , R ) C_ trCl ( z , A , R ) )
29	4	ad2antrr 762	. . . . . . 7 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) -> x e. A )
30		bnj1125 31060	. . . . . . 7 |- ( ( R FrSe A /\ x e. A /\ z e. trCl ( x , A , R ) ) -> trCl ( z , A , R ) C_ trCl ( x , A , R ) )
31	24, 29, 25, 30	syl3anc 1326	. . . . . 6 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) -> trCl ( z , A , R ) C_ trCl ( x , A , R ) )
32	28, 31	sstrd 3613	. . . . 5 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) -> pred ( z , A , R ) C_ trCl ( x , A , R ) )
33	32, 22	syl 17	. . . 4 |- ( ( ( ch /\ z e. E ) /\ z e. trCl ( x , A , R ) ) -> pred ( z , A , R ) C_ E )
34	1	bnj1424 30909	. . . . 5 |- ( z e. E -> ( z e. { x } \/ z e. trCl ( x , A , R ) ) )
35	34	adantl 482	. . . 4 |- ( ( ch /\ z e. E ) -> ( z e. { x } \/ z e. trCl ( x , A , R ) ) )
36	23, 33, 35	mpjaodan 827	. . 3 |- ( ( ch /\ z e. E ) -> pred ( z , A , R ) C_ E )
37	36	ralrimiva 2966	. 2 |- ( ch -> A. z e. E pred ( z , A , R ) C_ E )
38		snex 4908	. . . . . . . 8 |- { x } e. _V
39	38	a1i 11	. . . . . . 7 |- ( ch -> { x } e. _V )
40		bnj893 30998	. . . . . . . 8 |- ( ( R FrSe A /\ x e. A ) -> trCl ( x , A , R ) e. _V )
41	16, 4, 40	syl2anc 693	. . . . . . 7 |- ( ch -> trCl ( x , A , R ) e. _V )
42	39, 41	bnj1149 30863	. . . . . 6 |- ( ch -> ( { x } u. trCl ( x , A , R ) ) e. _V )
43	1, 42	syl5eqel 2705	. . . . 5 |- ( ch -> E e. _V )
44		bnj1452.1	. . . . . 6 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
45	44	bnj1454 30912	. . . . 5 |- ( E e. _V -> ( E e. B <-> [. E / d ]. ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) ) )
46	43, 45	syl 17	. . . 4 |- ( ch -> ( E e. B <-> [. E / d ]. ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) ) )
47		bnj602 30985	. . . . . . . 8 |- ( x = z -> pred ( x , A , R ) = pred ( z , A , R ) )
48	47	sseq1d 3632	. . . . . . 7 |- ( x = z -> ( pred ( x , A , R ) C_ d <-> pred ( z , A , R ) C_ d ) )
49	48	cbvralv 3171	. . . . . 6 |- ( A. x e. d pred ( x , A , R ) C_ d <-> A. z e. d pred ( z , A , R ) C_ d )
50	49	anbi2i 730	. . . . 5 |- ( ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) <-> ( d C_ A /\ A. z e. d pred ( z , A , R ) C_ d ) )
51	50	sbcbii 3491	. . . 4 |- ( [. E / d ]. ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) <-> [. E / d ]. ( d C_ A /\ A. z e. d pred ( z , A , R ) C_ d ) )
52	46, 51	syl6bb 276	. . 3 |- ( ch -> ( E e. B <-> [. E / d ]. ( d C_ A /\ A. z e. d pred ( z , A , R ) C_ d ) ) )
53		sseq1 3626	. . . . . 6 |- ( d = E -> ( d C_ A <-> E C_ A ) )
54		sseq2 3627	. . . . . . 7 |- ( d = E -> ( pred ( z , A , R ) C_ d <-> pred ( z , A , R ) C_ E ) )
55	54	raleqbi1dv 3146	. . . . . 6 |- ( d = E -> ( A. z e. d pred ( z , A , R ) C_ d <-> A. z e. E pred ( z , A , R ) C_ E ) )
56	53, 55	anbi12d 747	. . . . 5 |- ( d = E -> ( ( d C_ A /\ A. z e. d pred ( z , A , R ) C_ d ) <-> ( E C_ A /\ A. z e. E pred ( z , A , R ) C_ E ) ) )
57	56	sbcieg 3468	. . . 4 |- ( E e. _V -> ( [. E / d ]. ( d C_ A /\ A. z e. d pred ( z , A , R ) C_ d ) <-> ( E C_ A /\ A. z e. E pred ( z , A , R ) C_ E ) ) )
58	43, 57	syl 17	. . 3 |- ( ch -> ( [. E / d ]. ( d C_ A /\ A. z e. d pred ( z , A , R ) C_ d ) <-> ( E C_ A /\ A. z e. E pred ( z , A , R ) C_ E ) ) )
59	52, 58	bitrd 268	. 2 |- ( ch -> ( E e. B <-> ( E C_ A /\ A. z e. E pred ( z , A , R ) C_ E ) ) )
60	9, 37, 59	mpbir2and 957	1 |- ( ch -> E e. B )

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   _Vcvv 3200   [.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-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  df-bnj19 30763

This theorem is referenced by:  bnj1312  31126