Metamath Proof Explorer


Theorem bnj1398

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 bnj1398.1 B = d | d A x d pred x A R d
bnj1398.2 Y = x f pred x A R
bnj1398.3 C = f | d B f Fn d x d f x = G Y
bnj1398.4 τ f C dom f = x trCl x A R
bnj1398.5 D = x A | ¬ f τ
bnj1398.6 ψ R FrSe A D
bnj1398.7 χ ψ x D y D ¬ y R x
bnj1398.8 No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
bnj1398.9 No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
bnj1398.10 P = H
bnj1398.11 θ χ z y pred x A R y trCl y A R
bnj1398.12 η θ y pred x A R z y trCl y A R
Assertion bnj1398 χ y pred x A R y trCl y A R = dom P

Proof

Step Hyp Ref Expression
1 bnj1398.1 B = d | d A x d pred x A R d
2 bnj1398.2 Y = x f pred x A R
3 bnj1398.3 C = f | d B f Fn d x d f x = G Y
4 bnj1398.4 τ f C dom f = x trCl x A R
5 bnj1398.5 D = x A | ¬ f τ
6 bnj1398.6 ψ R FrSe A D
7 bnj1398.7 χ ψ x D y D ¬ y R x
8 bnj1398.8 Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9 bnj1398.9 Could not format H = { f | E. y e. _pred ( x , A , R ) ta' } : No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
10 bnj1398.10 P = H
11 bnj1398.11 θ χ z y pred x A R y trCl y A R
12 bnj1398.12 η θ y pred x A R z y trCl y A R
13 df-iun y pred x A R y trCl y A R = z | y pred x A R z y trCl y A R
14 13 bnj1436 z y pred x A R y trCl y A R y pred x A R z y trCl y A R
15 11 14 simplbiim θ y pred x A R z y trCl y A R
16 nfv y ψ
17 nfv y x D
18 nfra1 y y D ¬ y R x
19 16 17 18 nf3an y ψ x D y D ¬ y R x
20 7 19 nfxfr y χ
21 nfiu1 _ y y pred x A R y trCl y A R
22 21 nfcri y z y pred x A R y trCl y A R
23 20 22 nfan y χ z y pred x A R y trCl y A R
24 11 23 nfxfr y θ
25 24 nf5ri θ y θ
26 15 12 25 bnj1521 θ y η
27 nfv f R FrSe A
28 nfe1 f f τ
29 28 nfn f ¬ f τ
30 nfcv _ f A
31 29 30 nfrabw _ f x A | ¬ f τ
32 5 31 nfcxfr _ f D
33 nfcv _ f
34 32 33 nfne f D
35 27 34 nfan f R FrSe A D
36 6 35 nfxfr f ψ
37 32 nfcri f x D
38 nfv f ¬ y R x
39 32 38 nfralw f y D ¬ y R x
40 36 37 39 nf3an f ψ x D y D ¬ y R x
41 7 40 nfxfr f χ
42 nfv f z y pred x A R y trCl y A R
43 41 42 nfan f χ z y pred x A R y trCl y A R
44 11 43 nfxfr f θ
45 nfv f y pred x A R
46 nfv f z y trCl y A R
47 44 45 46 nf3an f θ y pred x A R z y trCl y A R
48 12 47 nfxfr f η
49 48 nf5ri η f η
50 11 simplbi θ χ
51 12 50 bnj835 η χ
52 12 simp2bi η y pred x A R
53 1 2 3 4 5 6 7 8 bnj1388 Could not format ( ch -> A. y e. _pred ( x , A , R ) E. f ta' ) : No typesetting found for |- ( ch -> A. y e. _pred ( x , A , R ) E. f ta' ) with typecode |-
54 rsp Could not format ( A. y e. _pred ( x , A , R ) E. f ta' -> ( y e. _pred ( x , A , R ) -> E. f ta' ) ) : No typesetting found for |- ( A. y e. _pred ( x , A , R ) E. f ta' -> ( y e. _pred ( x , A , R ) -> E. f ta' ) ) with typecode |-
55 53 54 syl Could not format ( ch -> ( y e. _pred ( x , A , R ) -> E. f ta' ) ) : No typesetting found for |- ( ch -> ( y e. _pred ( x , A , R ) -> E. f ta' ) ) with typecode |-
56 51 52 55 sylc Could not format ( et -> E. f ta' ) : No typesetting found for |- ( et -> E. f ta' ) with typecode |-
57 49 56 bnj596 Could not format ( et -> E. f ( et /\ ta' ) ) : No typesetting found for |- ( et -> E. f ( et /\ ta' ) ) with typecode |-
58 1 2 3 4 8 bnj1373 Could not format ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) : No typesetting found for |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) with typecode |-
59 58 simplbi Could not format ( ta' -> f e. C ) : No typesetting found for |- ( ta' -> f e. C ) with typecode |-
60 59 adantl Could not format ( ( et /\ ta' ) -> f e. C ) : No typesetting found for |- ( ( et /\ ta' ) -> f e. C ) with typecode |-
61 58 simprbi Could not format ( ta' -> dom f = ( { y } u. _trCl ( y , A , R ) ) ) : No typesetting found for |- ( ta' -> dom f = ( { y } u. _trCl ( y , A , R ) ) ) with typecode |-
62 rspe y pred x A R dom f = y trCl y A R y pred x A R dom f = y trCl y A R
63 52 61 62 syl2an Could not format ( ( et /\ ta' ) -> E. y e. _pred ( x , A , R ) dom f = ( { y } u. _trCl ( y , A , R ) ) ) : No typesetting found for |- ( ( et /\ ta' ) -> E. y e. _pred ( x , A , R ) dom f = ( { y } u. _trCl ( y , A , R ) ) ) with typecode |-
64 9 abeq2i Could not format ( f e. H <-> E. y e. _pred ( x , A , R ) ta' ) : No typesetting found for |- ( f e. H <-> E. y e. _pred ( x , A , R ) ta' ) with typecode |-
65 58 rexbii Could not format ( E. y e. _pred ( x , A , R ) ta' <-> E. y e. _pred ( x , A , R ) ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) : No typesetting found for |- ( E. y e. _pred ( x , A , R ) ta' <-> E. y e. _pred ( x , A , R ) ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) with typecode |-
66 r19.42v y pred x A R f C dom f = y trCl y A R f C y pred x A R dom f = y trCl y A R
67 64 65 66 3bitri f H f C y pred x A R dom f = y trCl y A R
68 60 63 67 sylanbrc Could not format ( ( et /\ ta' ) -> f e. H ) : No typesetting found for |- ( ( et /\ ta' ) -> f e. H ) with typecode |-
69 12 simp3bi η z y trCl y A R
70 69 adantr Could not format ( ( et /\ ta' ) -> z e. ( { y } u. _trCl ( y , A , R ) ) ) : No typesetting found for |- ( ( et /\ ta' ) -> z e. ( { y } u. _trCl ( y , A , R ) ) ) with typecode |-
71 61 adantl Could not format ( ( et /\ ta' ) -> dom f = ( { y } u. _trCl ( y , A , R ) ) ) : No typesetting found for |- ( ( et /\ ta' ) -> dom f = ( { y } u. _trCl ( y , A , R ) ) ) with typecode |-
72 70 71 eleqtrrd Could not format ( ( et /\ ta' ) -> z e. dom f ) : No typesetting found for |- ( ( et /\ ta' ) -> z e. dom f ) with typecode |-
73 68 72 jca Could not format ( ( et /\ ta' ) -> ( f e. H /\ z e. dom f ) ) : No typesetting found for |- ( ( et /\ ta' ) -> ( f e. H /\ z e. dom f ) ) with typecode |-
74 57 73 bnj593 η f f H z dom f
75 df-rex f H z dom f f f H z dom f
76 74 75 sylibr η f H z dom f
77 10 dmeqi dom P = dom H
78 9 bnj1317 w H f w H
79 78 bnj1400 dom H = f H dom f
80 77 79 eqtri dom P = f H dom f
81 80 eleq2i z dom P z f H dom f
82 eliun z f H dom f f H z dom f
83 81 82 bitri z dom P f H z dom f
84 76 83 sylibr η z dom P
85 26 84 bnj593 θ y z dom P
86 nfre1 Could not format F/ y E. y e. _pred ( x , A , R ) ta' : No typesetting found for |- F/ y E. y e. _pred ( x , A , R ) ta' with typecode |-
87 86 nfab Could not format F/_ y { f | E. y e. _pred ( x , A , R ) ta' } : No typesetting found for |- F/_ y { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
88 9 87 nfcxfr _ y H
89 88 nfuni _ y H
90 10 89 nfcxfr _ y P
91 90 nfdm _ y dom P
92 91 nfcrii z dom P y z dom P
93 85 92 bnj1397 θ z dom P
94 11 93 sylbir χ z y pred x A R y trCl y A R z dom P
95 94 ex χ z y pred x A R y trCl y A R z dom P
96 95 ssrdv χ y pred x A R y trCl y A R dom P
97 simpr f H z dom f z dom f
98 67 simprbi f H y pred x A R dom f = y trCl y A R
99 98 adantr f H z dom f y pred x A R dom f = y trCl y A R
100 r19.42v y pred x A R z dom f dom f = y trCl y A R z dom f y pred x A R dom f = y trCl y A R
101 eleq2 dom f = y trCl y A R z dom f z y trCl y A R
102 101 biimpac z dom f dom f = y trCl y A R z y trCl y A R
103 102 reximi y pred x A R z dom f dom f = y trCl y A R y pred x A R z y trCl y A R
104 100 103 sylbir z dom f y pred x A R dom f = y trCl y A R y pred x A R z y trCl y A R
105 97 99 104 syl2anc f H z dom f y pred x A R z y trCl y A R
106 105 rexlimiva f H z dom f y pred x A R z y trCl y A R
107 eliun z y pred x A R y trCl y A R y pred x A R z y trCl y A R
108 106 83 107 3imtr4i z dom P z y pred x A R y trCl y A R
109 108 ssriv dom P y pred x A R y trCl y A R
110 109 a1i χ dom P y pred x A R y trCl y A R
111 96 110 eqssd χ y pred x A R y trCl y A R = dom P