Metamath Proof Explorer

Theorem bnj1286

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 bnj1286.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1286.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1286.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1286.4 
|- D = ( dom g i^i dom h )
bnj1286.5 
|- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
bnj1286.6 
|- ( ph <-> ( R _FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
bnj1286.7 
|- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
Assertion bnj1286 
|- ( ps -> _pred ( x , A , R ) C_ D )

Proof

Step Hyp Ref Expression
1 bnj1286.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1286.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1286.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1286.4 
 |-  D = ( dom g i^i dom h )
5 bnj1286.5 
 |-  E = { x e. D | ( g ` x ) =/= ( h ` x ) }
6 bnj1286.6 
 |-  ( ph <-> ( R _FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
7 bnj1286.7 
 |-  ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
8 1 2 3 4 5 6 7 bnj1256 
 |-  ( ph -> E. d e. B g Fn d )
9 8 bnj1196 
 |-  ( ph -> E. d ( d e. B /\ g Fn d ) )
10 1 bnj1517 
 |-  ( d e. B -> A. x e. d _pred ( x , A , R ) C_ d )
11 10 adantr 
 |-  ( ( d e. B /\ g Fn d ) -> A. x e. d _pred ( x , A , R ) C_ d )
12 fndm 
 |-  ( g Fn d -> dom g = d )
13 sseq2 
 |-  ( dom g = d -> ( _pred ( x , A , R ) C_ dom g <-> _pred ( x , A , R ) C_ d ) )
14 13 raleqbi1dv 
 |-  ( dom g = d -> ( A. x e. dom g _pred ( x , A , R ) C_ dom g <-> A. x e. d _pred ( x , A , R ) C_ d ) )
15 12 14 syl 
 |-  ( g Fn d -> ( A. x e. dom g _pred ( x , A , R ) C_ dom g <-> A. x e. d _pred ( x , A , R ) C_ d ) )
16 15 adantl 
 |-  ( ( d e. B /\ g Fn d ) -> ( A. x e. dom g _pred ( x , A , R ) C_ dom g <-> A. x e. d _pred ( x , A , R ) C_ d ) )
17 11 16 mpbird 
 |-  ( ( d e. B /\ g Fn d ) -> A. x e. dom g _pred ( x , A , R ) C_ dom g )
18 9 17 bnj593 
 |-  ( ph -> E. d A. x e. dom g _pred ( x , A , R ) C_ dom g )
19 18 bnj937 
 |-  ( ph -> A. x e. dom g _pred ( x , A , R ) C_ dom g )
20 7 19 bnj835 
 |-  ( ps -> A. x e. dom g _pred ( x , A , R ) C_ dom g )
21 5 ssrab3 
 |-  E C_ D
22 4 bnj1292 
 |-  D C_ dom g
23 21 22 sstri 
 |-  E C_ dom g
24 23 sseli 
 |-  ( x e. E -> x e. dom g )
25 7 24 bnj836 
 |-  ( ps -> x e. dom g )
26 20 25 bnj1294 
 |-  ( ps -> _pred ( x , A , R ) C_ dom g )
27 1 2 3 4 5 6 7 bnj1259 
 |-  ( ph -> E. d e. B h Fn d )
28 27 bnj1196 
 |-  ( ph -> E. d ( d e. B /\ h Fn d ) )
29 10 adantr 
 |-  ( ( d e. B /\ h Fn d ) -> A. x e. d _pred ( x , A , R ) C_ d )
30 fndm 
 |-  ( h Fn d -> dom h = d )
31 sseq2 
 |-  ( dom h = d -> ( _pred ( x , A , R ) C_ dom h <-> _pred ( x , A , R ) C_ d ) )
32 31 raleqbi1dv 
 |-  ( dom h = d -> ( A. x e. dom h _pred ( x , A , R ) C_ dom h <-> A. x e. d _pred ( x , A , R ) C_ d ) )
33 30 32 syl 
 |-  ( h Fn d -> ( A. x e. dom h _pred ( x , A , R ) C_ dom h <-> A. x e. d _pred ( x , A , R ) C_ d ) )
34 33 adantl 
 |-  ( ( d e. B /\ h Fn d ) -> ( A. x e. dom h _pred ( x , A , R ) C_ dom h <-> A. x e. d _pred ( x , A , R ) C_ d ) )
35 29 34 mpbird 
 |-  ( ( d e. B /\ h Fn d ) -> A. x e. dom h _pred ( x , A , R ) C_ dom h )
36 28 35 bnj593 
 |-  ( ph -> E. d A. x e. dom h _pred ( x , A , R ) C_ dom h )
37 36 bnj937 
 |-  ( ph -> A. x e. dom h _pred ( x , A , R ) C_ dom h )
38 7 37 bnj835 
 |-  ( ps -> A. x e. dom h _pred ( x , A , R ) C_ dom h )
39 4 bnj1293 
 |-  D C_ dom h
40 21 39 sstri 
 |-  E C_ dom h
41 40 sseli 
 |-  ( x e. E -> x e. dom h )
42 7 41 bnj836 
 |-  ( ps -> x e. dom h )
43 38 42 bnj1294 
 |-  ( ps -> _pred ( x , A , R ) C_ dom h )
44 26 43 ssind 
 |-  ( ps -> _pred ( x , A , R ) C_ ( dom g i^i dom h ) )
45 44 4 sseqtrrdi 
 |-  ( ps -> _pred ( x , A , R ) C_ D )