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Metamath Proof Explorer

Theorem bnj1388

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 bnj1388.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1388.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1388.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1388.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1388.5 
|- D = { x e. A | -. E. f ta }
bnj1388.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1388.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1388.8 
|- ( ta' <-> [. y / x ]. ta )
Assertion bnj1388 
|- ( ch -> A. y e. _pred ( x , A , R ) E. f ta' )

Proof

Step Hyp Ref Expression
1 bnj1388.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1388.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1388.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1388.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1388.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1388.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1388.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1388.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 nfv 
 |-  F/ y ps
10 nfv 
 |-  F/ y x e. D
11 nfra1 
 |-  F/ y A. y e. D -. y R x
12 9 10 11 nf3an 
 |-  F/ y ( ps /\ x e. D /\ A. y e. D -. y R x )
13 7 12 nfxfr 
 |-  F/ y ch
14 bnj1152 
 |-  ( y e. _pred ( x , A , R ) <-> ( y e. A /\ y R x ) )
15 14 simplbi 
 |-  ( y e. _pred ( x , A , R ) -> y e. A )
16 15 adantl 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> y e. A )
17 14 biimpi 
 |-  ( y e. _pred ( x , A , R ) -> ( y e. A /\ y R x ) )
18 17 adantl 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> ( y e. A /\ y R x ) )
19 18 simprd 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> y R x )
20 7 simp3bi 
 |-  ( ch -> A. y e. D -. y R x )
21 20 adantr 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> A. y e. D -. y R x )
22 df-ral 
 |-  ( A. y e. D -. y R x <-> A. y ( y e. D -> -. y R x ) )
23 con2b 
 |-  ( ( y e. D -> -. y R x ) <-> ( y R x -> -. y e. D ) )
24 23 albii 
 |-  ( A. y ( y e. D -> -. y R x ) <-> A. y ( y R x -> -. y e. D ) )
25 22 24 bitri 
 |-  ( A. y e. D -. y R x <-> A. y ( y R x -> -. y e. D ) )
26 sp 
 |-  ( A. y ( y R x -> -. y e. D ) -> ( y R x -> -. y e. D ) )
27 26 impcom 
 |-  ( ( y R x /\ A. y ( y R x -> -. y e. D ) ) -> -. y e. D )
28 25 27 sylan2b 
 |-  ( ( y R x /\ A. y e. D -. y R x ) -> -. y e. D )
29 19 21 28 syl2anc 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> -. y e. D )
30 5 eleq2i 
 |-  ( y e. D <-> y e. { x e. A | -. E. f ta } )
31 nfcv 
 |-  F/_ x y
32 nfcv 
 |-  F/_ x A
33 nfsbc1v 
 |-  F/ x [. y / x ]. ta
34 8 33 nfxfr 
 |-  F/ x ta'
35 34 nfex 
 |-  F/ x E. f ta'
36 35 nfn 
 |-  F/ x -. E. f ta'
37 sbceq1a 
 |-  ( x = y -> ( ta <-> [. y / x ]. ta ) )
38 37 8 bitr4di 
 |-  ( x = y -> ( ta <-> ta' ) )
39 38 exbidv 
 |-  ( x = y -> ( E. f ta <-> E. f ta' ) )
40 39 notbid 
 |-  ( x = y -> ( -. E. f ta <-> -. E. f ta' ) )
41 31 32 36 40 elrabf 
 |-  ( y e. { x e. A | -. E. f ta } <-> ( y e. A /\ -. E. f ta' ) )
42 30 41 bitri 
 |-  ( y e. D <-> ( y e. A /\ -. E. f ta' ) )
43 29 42 sylnib 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> -. ( y e. A /\ -. E. f ta' ) )
44 iman 
 |-  ( ( y e. A -> E. f ta' ) <-> -. ( y e. A /\ -. E. f ta' ) )
45 43 44 sylibr 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> ( y e. A -> E. f ta' ) )
46 16 45 mpd 
 |-  ( ( ch /\ y e. _pred ( x , A , R ) ) -> E. f ta' )
47 46 ex 
 |-  ( ch -> ( y e. _pred ( x , A , R ) -> E. f ta' ) )
48 13 47 ralrimi 
 |-  ( ch -> A. y e. _pred ( x , A , R ) E. f ta' )