Theorem bnj1312 32401

Description: Technical lemma for bnj60 32405. 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 460	. . . . . . 7 |- ( ps -> R FrSe A )
4	1	bnj21 32058	. . . . . . . 8 |- D C_ A
5	4	a1i 11	. . . . . . 7 |- ( ps -> D C_ A )
6	2	simprbi 464	. . . . . . 7 |- ( ps -> D =/= (/) )
7	1	bnj1230 32148	. . . . . . . 8 |- ( w e. D -> A. x w e. D )
8	7	bnj1228 32354	. . . . . . 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 1219	. . . . . 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 1674	. . . . . . . . 9 |- F/ x R FrSe A
12	7	nfcii 2606	. . . . . . . . . 10 |- F/_ x D
13		nfcv 2616	. . . . . . . . . 10 |- F/_ x (/)
14	12, 13	nfne 2783	. . . . . . . . 9 |- F/ x D =/= (/)
15	11, 14	nfan 1866	. . . . . . . 8 |- F/ x ( R FrSe A /\ D =/= (/) )
16	2, 15	nfxfr 1616	. . . . . . 7 |- F/ x ps
17	16	nfri 1813	. . . . . 6 |- ( ps -> A. x ps )
18	9, 10, 17	bnj1521 32196	. . . . 5 |- ( ps -> E. x ch )
19	10	simp2bi 1004	. . . . 5 |- ( ch -> x e. D )
20	1	bnj1538 32200	. . . . . 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 32399	. . . . . . . 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 32104	. . . . . . . . . . . . . 14 |- ( ch -> R FrSe A )
34	21, 22, 23, 24, 1, 2, 10, 25, 26, 27	bnj1384 32375	. . . . . . . . . . . . . 14 |- ( R FrSe A -> Fun P )
35	33, 34	syl 16	. . . . . . . . . . . . 13 |- ( ch -> Fun P )
36	21, 22, 23, 24, 1, 2, 10, 25, 26, 27	bnj1415 32381	. . . . . . . . . . . . 13 |- ( ch -> dom P = trCl ( x , A , R ) )
37	35, 36	bnj1422 32183	. . . . . . . . . . . 12 |- ( ch -> P Fn trCl ( x , A , R ) )
38	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 36	bnj1416 32382	. . . . . . . . . . . . . 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 32385	. . . . . . . . . . . . 13 |- ( ch -> Fun Q )
40	39, 38	bnj1422 32183	. . . . . . . . . . . 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 32394	. . . . . . . . . . 11 |- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )
42	32	fneq2i 5617	. . . . . . . . . . . 12 |- ( Q Fn E <-> Q Fn ( { x } u. trCl ( x , A , R ) ) )
43	40, 42	sylibr 212	. . . . . . . . . . 11 |- ( ch -> Q Fn E )
44	21, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32	bnj1452 32395	. . . . . . . . . . 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 32398	. . . . . . . . . 10 |- ( ch -> Q e. C )
46	45, 38	jca 532	. . . . . . . . 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 32400	. . . . . . . 8 |- ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
48	30, 47	mpdan 668	. . . . . . 7 |- ( ch -> E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
49	48, 24	bnj1198 32141	. . . . . 6 |- ( ch -> E. f ta )
50	20, 49	nsyl3 119	. . . . 5 |- ( ch -> -. x e. D )
51	18, 19, 50	bnj1304 32165	. . . 4 |- -. ps
52	2, 51	bnj1541 32201	. . 3 |- ( R FrSe A -> D = (/) )
53	1, 52	bnj1476 32192	. 2 |- ( R FrSe A -> A. x e. A E. f ta )
54	24	exbii 1635	. . . 4 |- ( E. f ta <-> E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
55		df-rex 2805	. . . 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 252	. . 3 |- ( E. f ta <-> E. f e. C dom f = ( { x } u. trCl ( x , A , R ) ) )
57	56	ralbii 2839	. 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 196	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 184   /\ wa 369   /\ w3a 965   = wceq 1370   E.wex 1587   e. wcel 1758   {cab 2439   =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803   _Vcvv 3078   [.wsbc 3294   u. cun 3437   C_ wss 3439   (/)c0 3748   {csn 3988   <.cop 3994   U.cuni 4202   class class class wbr 4403   dom cdm 4951   |` cres 4953   Fun wfun 5523   Fn wfn 5524   ` cfv 5529   predc-bnj14 32028   FrSe w-bnj15 32032   trClc-bnj18 32034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7921  ax-inf2 7961

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-1o 7033  df-bnj17 32027  df-bnj14 32029  df-bnj13 32031  df-bnj15 32033  df-bnj18 32035  df-bnj19 32037

This theorem is referenced by:  bnj1493  32402