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Metamath Proof Explorer

Theorem bnj1312

Description: Technical lemma for bnj60 . 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.)

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

Proof

Step Hyp Ref Expression
1 bnj1312.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1312.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1312.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1312.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1312.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1312.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1312.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1312.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1312.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1312.10 
 |-  P = U. H
11 bnj1312.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1312.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1312.13 
 |-  W = <. z , ( Q |` _pred ( z , A , R ) ) >.
14 bnj1312.14 
 |-  E = ( { x } u. _trCl ( x , A , R ) )
15 6 simplbi 
 |-  ( ps -> R _FrSe A )
16 5 ssrab3 
 |-  D C_ A
17 16 a1i 
 |-  ( ps -> D C_ A )
18 6 simprbi 
 |-  ( ps -> D =/= (/) )
19 5 bnj1230 
 |-  ( w e. D -> A. x w e. D )
20 19 bnj1228 
 |-  ( ( R _FrSe A /\ D C_ A /\ D =/= (/) ) -> E. x e. D A. y e. D -. y R x )
21 15 17 18 20 syl3anc 
 |-  ( ps -> E. x e. D A. y e. D -. y R x )
22 nfv 
 |-  F/ x R _FrSe A
23 19 nfcii 
 |-  F/_ x D
24 nfcv 
 |-  F/_ x (/)
25 23 24 nfne 
 |-  F/ x D =/= (/)
26 22 25 nfan 
 |-  F/ x ( R _FrSe A /\ D =/= (/) )
27 6 26 nfxfr 
 |-  F/ x ps
28 27 nf5ri 
 |-  ( ps -> A. x ps )
29 21 7 28 bnj1521 
 |-  ( ps -> E. x ch )
30 7 simp2bi 
 |-  ( ch -> x e. D )
31 5 bnj1538 
 |-  ( x e. D -> -. E. f ta )
32 1 2 3 4 5 6 7 8 9 10 11 12 bnj1489 
 |-  ( ch -> Q e. _V )
33 7 15 bnj835 
 |-  ( ch -> R _FrSe A )
34 1 2 3 4 5 6 7 8 9 10 bnj1384 
 |-  ( R _FrSe A -> Fun P )
35 33 34 syl 
 |-  ( ch -> Fun P )
36 1 2 3 4 5 6 7 8 9 10 bnj1415 
 |-  ( ch -> dom P = _trCl ( x , A , R ) )
37 35 36 bnj1422 
 |-  ( ch -> P Fn _trCl ( x , A , R ) )
38 1 2 3 4 5 6 7 8 9 10 11 12 36 bnj1416 
 |-  ( ch -> dom Q = ( { x } u. _trCl ( x , A , R ) ) )
39 1 2 3 4 5 6 7 8 9 10 11 12 35 38 36 bnj1421 
 |-  ( ch -> Fun Q )
40 39 38 bnj1422 
 |-  ( ch -> Q Fn ( { x } u. _trCl ( x , A , R ) ) )
41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 37 40 bnj1423 
 |-  ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )
42 14 fneq2i 
 |-  ( Q Fn E <-> Q Fn ( { x } u. _trCl ( x , A , R ) ) )
43 40 42 sylibr 
 |-  ( ch -> Q Fn E )
44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 bnj1452 
 |-  ( ch -> E e. B )
45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 32 41 43 44 bnj1463 
 |-  ( ch -> Q e. C )
46 45 38 jca 
 |-  ( ch -> ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) )
47 1 2 3 4 5 6 7 8 9 10 11 12 46 bnj1491 
 |-  ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
48 32 47 mpdan 
 |-  ( ch -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
49 48 4 bnj1198 
 |-  ( ch -> E. f ta )
50 31 49 nsyl3 
 |-  ( ch -> -. x e. D )
51 29 30 50 bnj1304 
 |-  -. ps
52 6 51 bnj1541 
 |-  ( R _FrSe A -> D = (/) )
53 5 52 bnj1476 
 |-  ( R _FrSe A -> A. x e. A E. f ta )
54 4 exbii 
 |-  ( E. f ta <-> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
55 df-rex 
 |-  ( 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 
 |-  ( E. f ta <-> E. f e. C dom f = ( { x } u. _trCl ( x , A , R ) ) )
57 56 ralbii 
 |-  ( 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 
 |-  ( R _FrSe A -> A. x e. A E. f e. C dom f = ( { x } u. _trCl ( x , A , R ) ) )