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Metamath Proof Explorer

Theorem bnj1416

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 bnj1416.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
bnj1416.2 ⊢ Y = x f ↾ pred x A R
bnj1416.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
bnj1416.4 ⊢ τ ↔ f ∈ C ∧ dom ⁡ f = x ∪ trCl x A R
bnj1416.5 ⊢ D = x ∈ A | ¬ ∃ f τ
bnj1416.6 ⊢ ψ ↔ R FrSe A ∧ D ≠ ∅
bnj1416.7 ⊢ χ ↔ ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
bnj1416.8 No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
bnj1416.9 No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
bnj1416.10 ⊢ P = ⋃ H
bnj1416.11 ⊢ Z = x P ↾ pred x A R
bnj1416.12 ⊢ Q = P ∪ x G ⁡ Z
bnj1416.28 ⊢ χ → dom ⁡ P = trCl x A R
Assertion bnj1416 ⊢ χ → dom ⁡ Q = x ∪ trCl x A R

Proof

Step Hyp Ref Expression
1 bnj1416.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
2 bnj1416.2 ⊢ Y = x f ↾ pred x A R
3 bnj1416.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
4 bnj1416.4 ⊢ τ ↔ f ∈ C ∧ dom ⁡ f = x ∪ trCl x A R
5 bnj1416.5 ⊢ D = x ∈ A | ¬ ∃ f τ
6 bnj1416.6 ⊢ ψ ↔ R FrSe A ∧ D ≠ ∅
7 bnj1416.7 ⊢ χ ↔ ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
8 bnj1416.8 Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9 bnj1416.9 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 bnj1416.10 ⊢ P = ⋃ H
11 bnj1416.11 ⊢ Z = x P ↾ pred x A R
12 bnj1416.12 ⊢ Q = P ∪ x G ⁡ Z
13 bnj1416.28 ⊢ χ → dom ⁡ P = trCl x A R
14 12 dmeqi ⊢ dom ⁡ Q = dom ⁡ P ∪ x G ⁡ Z
15 dmun ⊢ dom ⁡ P ∪ x G ⁡ Z = dom ⁡ P ∪ dom ⁡ x G ⁡ Z
16 fvex ⊢ G ⁡ Z ∈ V
17 16 dmsnop ⊢ dom ⁡ x G ⁡ Z = x
18 17 uneq2i ⊢ dom ⁡ P ∪ dom ⁡ x G ⁡ Z = dom ⁡ P ∪ x
19 14 15 18 3eqtri ⊢ dom ⁡ Q = dom ⁡ P ∪ x
20 13 uneq1d ⊢ χ → dom ⁡ P ∪ x = trCl x A R ∪ x
21 uncom ⊢ trCl x A R ∪ x = x ∪ trCl x A R
22 20 21 eqtrdi ⊢ χ → dom ⁡ P ∪ x = x ∪ trCl x A R
23 19 22 eqtrid ⊢ χ → dom ⁡ Q = x ∪ trCl x A R