Metamath Proof Explorer

Theorem bnj1489

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 bnj1489.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1489.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1489.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1489.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1489.5 
|- D = { x e. A | -. E. f ta }
bnj1489.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1489.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1489.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1489.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1489.10 
|- P = U. H
bnj1489.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1489.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
Assertion bnj1489 
|- ( ch -> Q e. _V )

Proof

Step Hyp Ref Expression
1 bnj1489.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1489.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1489.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1489.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1489.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1489.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1489.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1489.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1489.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1489.10 
 |-  P = U. H
11 bnj1489.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1489.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1364 
 |-  ( R _FrSe A -> R _Se A )
14 df-bnj13 
 |-  ( R _Se A <-> A. x e. A _pred ( x , A , R ) e. _V )
15 13 14 sylib 
 |-  ( R _FrSe A -> A. x e. A _pred ( x , A , R ) e. _V )
16 6 15 bnj832 
 |-  ( ps -> A. x e. A _pred ( x , A , R ) e. _V )
17 7 16 bnj835 
 |-  ( ch -> A. x e. A _pred ( x , A , R ) e. _V )
18 5 7 bnj1212 
 |-  ( ch -> x e. A )
19 17 18 bnj1294 
 |-  ( ch -> _pred ( x , A , R ) e. _V )
20 nfv 
 |-  F/ y ps
21 nfv 
 |-  F/ y x e. D
22 nfra1 
 |-  F/ y A. y e. D -. y R x
23 20 21 22 nf3an 
 |-  F/ y ( ps /\ x e. D /\ A. y e. D -. y R x )
24 7 23 nfxfr 
 |-  F/ y ch
25 6 simplbi 
 |-  ( ps -> R _FrSe A )
26 7 25 bnj835 
 |-  ( ch -> R _FrSe A )
27 26 adantr 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> R _FrSe A )
28 1 2 3 4 5 6 7 8 bnj1388 
 |-  ( ch -> A. y e. _pred ( x , A , R ) E. f ta' )
29 28 r19.21bi 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> E. f ta' )
30 nfv 
 |-  F/ x R _FrSe A
31 nfsbc1v 
 |-  F/ x [. y / x ]. ta
32 8 31 nfxfr 
 |-  F/ x ta'
33 32 nfex 
 |-  F/ x E. f ta'
34 30 33 nfan 
 |-  F/ x ( R _FrSe A /\ E. f ta' )
35 32 nfeuw 
 |-  F/ x E! f ta'
36 34 35 nfim 
 |-  F/ x ( ( R _FrSe A /\ E. f ta' ) -> E! f ta' )
37 sneq 
 |-  ( x = y -> { x } = { y } )
38 bnj1318 
 |-  ( x = y -> _trCl ( x , A , R ) = _trCl ( y , A , R ) )
39 37 38 uneq12d 
 |-  ( x = y -> ( { x } u. _trCl ( x , A , R ) ) = ( { y } u. _trCl ( y , A , R ) ) )
40 39 eqeq2d 
 |-  ( x = y -> ( dom f = ( { x } u. _trCl ( x , A , R ) ) <-> dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
41 40 anbi2d 
 |-  ( x = y -> ( ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) )
42 1 2 3 4 8 bnj1373 
 |-  ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
43 41 42 bitr4di 
 |-  ( x = y -> ( ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) <-> ta' ) )
44 43 exbidv 
 |-  ( x = y -> ( E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) <-> E. f ta' ) )
45 44 anbi2d 
 |-  ( x = y -> ( ( R _FrSe A /\ E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) <-> ( R _FrSe A /\ E. f ta' ) ) )
46 43 eubidv 
 |-  ( x = y -> ( E! f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) <-> E! f ta' ) )
47 45 46 imbi12d 
 |-  ( x = y -> ( ( ( R _FrSe A /\ E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) -> E! f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) <-> ( ( R _FrSe A /\ E. f ta' ) -> E! f ta' ) ) )
48 biid 
 |-  ( ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
49 1 2 3 48 bnj1321 
 |-  ( ( R _FrSe A /\ E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) -> E! f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
50 36 47 49 chvarfv 
 |-  ( ( R _FrSe A /\ E. f ta' ) -> E! f ta' )
51 27 29 50 syl2anc 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> E! f ta' )
52 51 ex 
 |-  ( ch -> ( y e. _pred ( x , A , R ) -> E! f ta' ) )
53 24 52 ralrimi 
 |-  ( ch -> A. y e. _pred ( x , A , R ) E! f ta' )
54 9 a1i 
 |-  ( ch -> H = { f | E. y e. _pred ( x , A , R ) ta' } )
55 biid 
 |-  ( ( _pred ( x , A , R ) e. _V /\ A. y e. _pred ( x , A , R ) E! f ta' /\ H = { f | E. y e. _pred ( x , A , R ) ta' } ) <-> ( _pred ( x , A , R ) e. _V /\ A. y e. _pred ( x , A , R ) E! f ta' /\ H = { f | E. y e. _pred ( x , A , R ) ta' } ) )
56 55 bnj1366 
 |-  ( ( _pred ( x , A , R ) e. _V /\ A. y e. _pred ( x , A , R ) E! f ta' /\ H = { f | E. y e. _pred ( x , A , R ) ta' } ) -> H e. _V )
57 19 53 54 56 syl3anc 
 |-  ( ch -> H e. _V )
58 57 uniexd 
 |-  ( ch -> U. H e. _V )
59 10 58 eqeltrid 
 |-  ( ch -> P e. _V )
60 snex 
 |-  { <. x , ( G ` Z ) >. } e. _V
61 60 a1i 
 |-  ( ch -> { <. x , ( G ` Z ) >. } e. _V )
62 59 61 bnj1149 
 |-  ( ch -> ( P u. { <. x , ( G ` Z ) >. } ) e. _V )
63 12 62 eqeltrid 
 |-  ( ch -> Q e. _V )