Metamath Proof Explorer


Theorem bnj1447

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 bnj1447.1 B = d | d A x d pred x A R d
bnj1447.2 Y = x f pred x A R
bnj1447.3 C = f | d B f Fn d x d f x = G Y
bnj1447.4 τ f C dom f = x trCl x A R
bnj1447.5 D = x A | ¬ f τ
bnj1447.6 ψ R FrSe A D
bnj1447.7 χ ψ x D y D ¬ y R x
bnj1447.8 No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
bnj1447.9 No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
bnj1447.10 P = H
bnj1447.11 Z = x P pred x A R
bnj1447.12 Q = P x G Z
bnj1447.13 W = z Q pred z A R
Assertion bnj1447 Q z = G W y Q z = G W

Proof

Step Hyp Ref Expression
1 bnj1447.1 B = d | d A x d pred x A R d
2 bnj1447.2 Y = x f pred x A R
3 bnj1447.3 C = f | d B f Fn d x d f x = G Y
4 bnj1447.4 τ f C dom f = x trCl x A R
5 bnj1447.5 D = x A | ¬ f τ
6 bnj1447.6 ψ R FrSe A D
7 bnj1447.7 χ ψ x D y D ¬ y R x
8 bnj1447.8 Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9 bnj1447.9 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 bnj1447.10 P = H
11 bnj1447.11 Z = x P pred x A R
12 bnj1447.12 Q = P x G Z
13 bnj1447.13 W = z Q pred z A R
14 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 |-
15 14 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 |-
16 9 15 nfcxfr _ y H
17 16 nfuni _ y H
18 10 17 nfcxfr _ y P
19 nfcv _ y x
20 nfcv _ y G
21 nfcv _ y pred x A R
22 18 21 nfres _ y P pred x A R
23 19 22 nfop _ y x P pred x A R
24 11 23 nfcxfr _ y Z
25 20 24 nffv _ y G Z
26 19 25 nfop _ y x G Z
27 26 nfsn _ y x G Z
28 18 27 nfun _ y P x G Z
29 12 28 nfcxfr _ y Q
30 nfcv _ y z
31 29 30 nffv _ y Q z
32 nfcv _ y pred z A R
33 29 32 nfres _ y Q pred z A R
34 30 33 nfop _ y z Q pred z A R
35 13 34 nfcxfr _ y W
36 20 35 nffv _ y G W
37 31 36 nfeq y Q z = G W
38 37 nf5ri Q z = G W y Q z = G W