Theorem bnj1442 31117

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
bnj1442.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1442.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1442.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1442.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1442.5	|- D = { x e. A | -. E. f ta }
bnj1442.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1442.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1442.8	|- ( ta' <-> [. y / x ]. ta )
bnj1442.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1442.10	|- P = U. H
bnj1442.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1442.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1442.13	|- W = <. z , ( Q |` pred ( z , A , R ) ) >.
bnj1442.14	|- E = ( { x } u. trCl ( x , A , R ) )
bnj1442.15	|- ( ch -> P Fn trCl ( x , A , R ) )
bnj1442.16	|- ( ch -> Q Fn ( { x } u. trCl ( x , A , R ) ) )
bnj1442.17	|- ( th <-> ( ch /\ z e. E ) )
bnj1442.18	|- ( et <-> ( th /\ z e. { x } ) )

Assertion

Ref	Expression
bnj1442	|- ( et -> ( Q ` z ) = ( G ` W ) )

Distinct variable group:   x, A

Allowed substitution hints:   ps( x, y, z, f, d)   ch( x, y, z, f, d)   th( x, y, z, f, d)   ta( x, y, z, f, d)   et( x, y, z, f, d)   A( y, z, f, d)   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( x, y, z, f, d)   E( x, y, z, f, d)   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 bnj1442

Step	Hyp	Ref	Expression
1		bnj1442.18	. . 3 |- ( et <-> ( th /\ z e. { x } ) )
2		bnj1442.17	. . . 4 |- ( th <-> ( ch /\ z e. E ) )
3		bnj1442.16	. . . . . 6 |- ( ch -> Q Fn ( { x } u. trCl ( x , A , R ) ) )
4	3	bnj930 30840	. . . . 5 |- ( ch -> Fun Q )
5		opex 4932	. . . . . . . 8 |- <. x , ( G ` Z ) >. e. _V
6	5	snid 4208	. . . . . . 7 |- <. x , ( G ` Z ) >. e. { <. x , ( G ` Z ) >. }
7		elun2 3781	. . . . . . 7 |- ( <. x , ( G ` Z ) >. e. { <. x , ( G ` Z ) >. } -> <. x , ( G ` Z ) >. e. ( P u. { <. x , ( G ` Z ) >. } ) )
8	6, 7	ax-mp 5	. . . . . 6 |- <. x , ( G ` Z ) >. e. ( P u. { <. x , ( G ` Z ) >. } )
9		bnj1442.12	. . . . . 6 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
10	8, 9	eleqtrri 2700	. . . . 5 |- <. x , ( G ` Z ) >. e. Q
11		funopfv 6235	. . . . 5 |- ( Fun Q -> ( <. x , ( G ` Z ) >. e. Q -> ( Q ` x ) = ( G ` Z ) ) )
12	4, 10, 11	mpisyl 21	. . . 4 |- ( ch -> ( Q ` x ) = ( G ` Z ) )
13	2, 12	bnj832 30828	. . 3 |- ( th -> ( Q ` x ) = ( G ` Z ) )
14	1, 13	bnj832 30828	. 2 |- ( et -> ( Q ` x ) = ( G ` Z ) )
15		elsni 4194	. . . 4 |- ( z e. { x } -> z = x )
16	1, 15	simplbiim 659	. . 3 |- ( et -> z = x )
17	16	fveq2d 6195	. 2 |- ( et -> ( Q ` z ) = ( Q ` x ) )
18		bnj602 30985	. . . . . . . 8 |- ( z = x -> pred ( z , A , R ) = pred ( x , A , R ) )
19	18	reseq2d 5396	. . . . . . 7 |- ( z = x -> ( Q |` pred ( z , A , R ) ) = ( Q |` pred ( x , A , R ) ) )
20	16, 19	syl 17	. . . . . 6 |- ( et -> ( Q |` pred ( z , A , R ) ) = ( Q |` pred ( x , A , R ) ) )
21	9	bnj931 30841	. . . . . . . . . 10 |- P C_ Q
22	21	a1i 11	. . . . . . . . 9 |- ( ch -> P C_ Q )
23		bnj1442.7	. . . . . . . . . . . 12 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
24		bnj1442.6	. . . . . . . . . . . . 13 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
25	24	simplbi 476	. . . . . . . . . . . 12 |- ( ps -> R FrSe A )
26	23, 25	bnj835 30829	. . . . . . . . . . 11 |- ( ch -> R FrSe A )
27		bnj1442.5	. . . . . . . . . . . 12 |- D = { x e. A | -. E. f ta }
28	27, 23	bnj1212 30870	. . . . . . . . . . 11 |- ( ch -> x e. A )
29		bnj906 31000	. . . . . . . . . . 11 |- ( ( R FrSe A /\ x e. A ) -> pred ( x , A , R ) C_ trCl ( x , A , R ) )
30	26, 28, 29	syl2anc 693	. . . . . . . . . 10 |- ( ch -> pred ( x , A , R ) C_ trCl ( x , A , R ) )
31		bnj1442.15	. . . . . . . . . . 11 |- ( ch -> P Fn trCl ( x , A , R ) )
32		fndm 5990	. . . . . . . . . . 11 |- ( P Fn trCl ( x , A , R ) -> dom P = trCl ( x , A , R ) )
33	31, 32	syl 17	. . . . . . . . . 10 |- ( ch -> dom P = trCl ( x , A , R ) )
34	30, 33	sseqtr4d 3642	. . . . . . . . 9 |- ( ch -> pred ( x , A , R ) C_ dom P )
35	4, 22, 34	bnj1503 30919	. . . . . . . 8 |- ( ch -> ( Q |` pred ( x , A , R ) ) = ( P |` pred ( x , A , R ) ) )
36	2, 35	bnj832 30828	. . . . . . 7 |- ( th -> ( Q |` pred ( x , A , R ) ) = ( P |` pred ( x , A , R ) ) )
37	1, 36	bnj832 30828	. . . . . 6 |- ( et -> ( Q |` pred ( x , A , R ) ) = ( P |` pred ( x , A , R ) ) )
38	20, 37	eqtrd 2656	. . . . 5 |- ( et -> ( Q |` pred ( z , A , R ) ) = ( P |` pred ( x , A , R ) ) )
39	16, 38	opeq12d 4410	. . . 4 |- ( et -> <. z , ( Q |` pred ( z , A , R ) ) >. = <. x , ( P |` pred ( x , A , R ) ) >. )
40		bnj1442.13	. . . 4 |- W = <. z , ( Q |` pred ( z , A , R ) ) >.
41		bnj1442.11	. . . 4 |- Z = <. x , ( P |` pred ( x , A , R ) ) >.
42	39, 40, 41	3eqtr4g 2681	. . 3 |- ( et -> W = Z )
43	42	fveq2d 6195	. 2 |- ( et -> ( G ` W ) = ( G ` Z ) )
44	14, 17, 43	3eqtr4d 2666	1 |- ( et -> ( Q ` z ) = ( G ` W ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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   Fun wfun 5882   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

This theorem is referenced by:  bnj1423  31119