Theorem bnj1312 31126

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
bnj1312.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1312.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1312.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1312.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1312.5	|- D = { x e. A | -. E. f ta }
bnj1312.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1312.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1312.8	|- ( ta' <-> [. y / x ]. ta )
bnj1312.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1312.10	|- P = U. H
bnj1312.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1312.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1312.13	|- W = <. z , ( Q |` pred ( z , A , R ) ) >.
bnj1312.14	|- E = ( { x } u. trCl ( x , A , R ) )

Assertion

Ref	Expression
bnj1312	|- ( R FrSe A -> A. x e. A E. f e. C dom f = ( { x } u. trCl ( x , A , R ) ) )

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

Allowed substitution hints:   ps( x, z, f, d)   ch( x, y, f, d)   ta( x, z, f, d)   B( x, y, z, d)   C( x, z, f, d)   D( x, z, f, d)   P( x, y, z, f, d)   Q( x, y, f, d)   E( x)   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 bnj1312

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1312.5	. . 3 |- D = { x e. A | -. E. f ta }
2		bnj1312.6	. . . 4 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
3	2	simplbi 476	. . . . . . 7 |- ( ps -> R FrSe A )
4	1	ssrab3 3688	. . . . . . . 8 |- D C_ A
5	4	a1i 11	. . . . . . 7 |- ( ps -> D C_ A )
6	2	simprbi 480	. . . . . . 7 |- ( ps -> D =/= (/) )
7	1	bnj1230 30873	. . . . . . . 8 |- ( w e. D -> A. x w e. D )
8	7	bnj1228 31079	. . . . . . 7 |- ( ( R FrSe A /\ D C_ A /\ D =/= (/) ) -> E. x e. D A. y e. D -. y R x )
9	3, 5, 6, 8	syl3anc 1326	. . . . . 6 |- ( ps -> E. x e. D A. y e. D -. y R x )
10		bnj1312.7	. . . . . 6 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
11		nfv 1843	. . . . . . . . 9 |- F/ x R FrSe A
12	7	nfcii 2755	. . . . . . . . . 10 |- F/_ x D
13		nfcv 2764	. . . . . . . . . 10 |- F/_ x (/)
14	12, 13	nfne 2894	. . . . . . . . 9 |- F/ x D =/= (/)
15	11, 14	nfan 1828	. . . . . . . 8 |- F/ x ( R FrSe A /\ D =/= (/) )
16	2, 15	nfxfr 1779	. . . . . . 7 |- F/ x ps
17	16	nf5ri 2065	. . . . . 6 |- ( ps -> A. x ps )
18	9, 10, 17	bnj1521 30921	. . . . 5 |- ( ps -> E. x ch )
19	10	simp2bi 1077	. . . . 5 |- ( ch -> x e. D )
20	1	bnj1538 30925	. . . . . 6 |- ( x e. D -> -. E. f ta )
21		bnj1312.1	. . . . . . . . 9 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
22		bnj1312.2	. . . . . . . . 9 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
23		bnj1312.3	. . . . . . . . 9 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
24		bnj1312.4	. . . . . . . . 9 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
25		bnj1312.8	. . . . . . . . 9 |- ( ta' <-> [. y / x ]. ta )
26		bnj1312.9	. . . . . . . . 9 |- H = { f | E. y e. pred ( x , A , R ) ta' }
27		bnj1312.10	. . . . . . . . 9 |- P = U. H
28		bnj1312.11	. . . . . . . . 9 |- Z = <. x , ( P |` pred ( x , A , R ) ) >.
29		bnj1312.12	. . . . . . . . 9 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
30	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29	bnj1489 31124	. . . . . . . 8 |- ( ch -> Q e. _V )
31		bnj1312.13	. . . . . . . . . . 11 |- W = <. z , ( Q |` pred ( z , A , R ) ) >.
32		bnj1312.14	. . . . . . . . . . 11 |- E = ( { x } u. trCl ( x , A , R ) )
33	10, 3	bnj835 30829	. . . . . . . . . . . . . 14 |- ( ch -> R FrSe A )
34	21, 22, 23, 24, 1, 2, 10, 25, 26, 27	bnj1384 31100	. . . . . . . . . . . . . 14 |- ( R FrSe A -> Fun P )
35	33, 34	syl 17	. . . . . . . . . . . . 13 |- ( ch -> Fun P )
36	21, 22, 23, 24, 1, 2, 10, 25, 26, 27	bnj1415 31106	. . . . . . . . . . . . 13 |- ( ch -> dom P = trCl ( x , A , R ) )
37	35, 36	bnj1422 30908	. . . . . . . . . . . 12 |- ( ch -> P Fn trCl ( x , A , R ) )
38	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 36	bnj1416 31107	. . . . . . . . . . . . . 14 |- ( ch -> dom Q = ( { x } u. trCl ( x , A , R ) ) )
39	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 35, 38, 36	bnj1421 31110	. . . . . . . . . . . . 13 |- ( ch -> Fun Q )
40	39, 38	bnj1422 30908	. . . . . . . . . . . 12 |- ( ch -> Q Fn ( { x } u. trCl ( x , A , R ) ) )
41	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 37, 40	bnj1423 31119	. . . . . . . . . . 11 |- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )
42	32	fneq2i 5986	. . . . . . . . . . . 12 |- ( Q Fn E <-> Q Fn ( { x } u. trCl ( x , A , R ) ) )
43	40, 42	sylibr 224	. . . . . . . . . . 11 |- ( ch -> Q Fn E )
44	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32	bnj1452 31120	. . . . . . . . . . 11 |- ( ch -> E e. B )
45	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 30, 41, 43, 44	bnj1463 31123	. . . . . . . . . 10 |- ( ch -> Q e. C )
46	45, 38	jca 554	. . . . . . . . 9 |- ( ch -> ( Q e. C /\ dom Q = ( { x } u. trCl ( x , A , R ) ) ) )
47	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 46	bnj1491 31125	. . . . . . . 8 |- ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
48	30, 47	mpdan 702	. . . . . . 7 |- ( ch -> E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
49	48, 24	bnj1198 30866	. . . . . 6 |- ( ch -> E. f ta )
50	20, 49	nsyl3 133	. . . . 5 |- ( ch -> -. x e. D )
51	18, 19, 50	bnj1304 30890	. . . 4 |- -. ps
52	2, 51	bnj1541 30926	. . 3 |- ( R FrSe A -> D = (/) )
53	1, 52	bnj1476 30917	. 2 |- ( R FrSe A -> A. x e. A E. f ta )
54	24	exbii 1774	. . . 4 |- ( E. f ta <-> E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
55		df-rex 2918	. . . 4 |- ( E. f e. C dom f = ( { x } u. trCl ( x , A , R ) ) <-> E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
56	54, 55	bitr4i 267	. . 3 |- ( E. f ta <-> E. f e. C dom f = ( { x } u. trCl ( x , A , R ) ) )
57	56	ralbii 2980	. 2 |- ( A. x e. A E. f ta <-> A. x e. A E. f e. C dom f = ( { x } u. trCl ( x , A , R ) ) )
58	53, 57	sylib 208	1 |- ( R FrSe A -> A. x e. A E. f e. C dom f = ( { x } u. trCl ( x , A , R ) ) )

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   _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   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  df-bnj19 30763

This theorem is referenced by:  bnj1493  31127