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Metamath Proof Explorer

Theorem bnj1466

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 bnj1466.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
bnj1466.2 ⊢ Y = x f ↾ pred x A R
bnj1466.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
bnj1466.4 ⊢ τ ↔ f ∈ C ∧ dom ⁡ f = x ∪ trCl x A R
bnj1466.5 ⊢ D = x ∈ A | ¬ ∃ f τ
bnj1466.6 ⊢ ψ ↔ R FrSe A ∧ D ≠ ∅
bnj1466.7 ⊢ χ ↔ ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
bnj1466.8 No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
bnj1466.9 No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
bnj1466.10 ⊢ P = ⋃ H
bnj1466.11 ⊢ Z = x P ↾ pred x A R
bnj1466.12 ⊢ Q = P ∪ x G ⁡ Z
Assertion bnj1466 ⊢ w ∈ Q → ∀ f w ∈ Q

Proof

Step Hyp Ref Expression
1 bnj1466.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
2 bnj1466.2 ⊢ Y = x f ↾ pred x A R
3 bnj1466.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
4 bnj1466.4 ⊢ τ ↔ f ∈ C ∧ dom ⁡ f = x ∪ trCl x A R
5 bnj1466.5 ⊢ D = x ∈ A | ¬ ∃ f τ
6 bnj1466.6 ⊢ ψ ↔ R FrSe A ∧ D ≠ ∅
7 bnj1466.7 ⊢ χ ↔ ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
8 bnj1466.8 Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9 bnj1466.9 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 bnj1466.10 ⊢ P = ⋃ H
11 bnj1466.11 ⊢ Z = x P ↾ pred x A R
12 bnj1466.12 ⊢ Q = P ∪ x G ⁡ Z
13 9 bnj1317 ⊢ w ∈ H → ∀ f w ∈ H
14 13 nfcii ⊢ Ⅎ _ f H
15 14 nfuni ⊢ Ⅎ _ f ⋃ H
16 10 15 nfcxfr ⊢ Ⅎ _ f P
17 nfcv ⊢ Ⅎ _ f x
18 nfcv ⊢ Ⅎ _ f G
19 nfcv ⊢ Ⅎ _ f pred x A R
20 16 19 nfres ⊢ Ⅎ _ f P ↾ pred x A R
21 17 20 nfop ⊢ Ⅎ _ f x P ↾ pred x A R
22 11 21 nfcxfr ⊢ Ⅎ _ f Z
23 18 22 nffv ⊢ Ⅎ _ f G ⁡ Z
24 17 23 nfop ⊢ Ⅎ _ f x G ⁡ Z
25 24 nfsn ⊢ Ⅎ _ f x G ⁡ Z
26 16 25 nfun ⊢ Ⅎ _ f P ∪ x G ⁡ Z
27 12 26 nfcxfr ⊢ Ⅎ _ f Q
28 27 nfcrii ⊢ w ∈ Q → ∀ f w ∈ Q