Metamath Proof Explorer

Theorem bnj1259

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 bnj1259.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
bnj1259.2 ⊢ Y = x f ↾ pred x A R
bnj1259.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
bnj1259.4 ⊢ D = dom ⁡ g ∩ dom ⁡ h
bnj1259.5 ⊢ E = x ∈ D | g ⁡ x ≠ h ⁡ x
bnj1259.6 ⊢ φ ↔ R FrSe A ∧ g ∈ C ∧ h ∈ C ∧ g ↾ D ≠ h ↾ D
bnj1259.7 ⊢ ψ ↔ φ ∧ x ∈ E ∧ ∀ y ∈ E ¬ y R x
Assertion bnj1259 ⊢ φ → ∃ d ∈ B h Fn d

Proof

Step Hyp Ref Expression
1 bnj1259.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
2 bnj1259.2 ⊢ Y = x f ↾ pred x A R
3 bnj1259.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
4 bnj1259.4 ⊢ D = dom ⁡ g ∩ dom ⁡ h
5 bnj1259.5 ⊢ E = x ∈ D | g ⁡ x ≠ h ⁡ x
6 bnj1259.6 ⊢ φ ↔ R FrSe A ∧ g ∈ C ∧ h ∈ C ∧ g ↾ D ≠ h ↾ D
7 bnj1259.7 ⊢ ψ ↔ φ ∧ x ∈ E ∧ ∀ y ∈ E ¬ y R x
8 abid ⊢ h ∈ h | ∃ d ∈ B h Fn d ∧ ∀ x ∈ d h ⁡ x = G ⁡ x h ↾ pred x A R ↔ ∃ d ∈ B h Fn d ∧ ∀ x ∈ d h ⁡ x = G ⁡ x h ↾ pred x A R
9 8 bnj1238 ⊢ h ∈ h | ∃ d ∈ B h Fn d ∧ ∀ x ∈ d h ⁡ x = G ⁡ x h ↾ pred x A R → ∃ d ∈ B h Fn d
10 eqid ⊢ x h ↾ pred x A R = x h ↾ pred x A R
11 eqid ⊢ h | ∃ d ∈ B h Fn d ∧ ∀ x ∈ d h ⁡ x = G ⁡ x h ↾ pred x A R = h | ∃ d ∈ B h Fn d ∧ ∀ x ∈ d h ⁡ x = G ⁡ x h ↾ pred x A R
12 2 3 10 11 bnj1234 ⊢ C = h | ∃ d ∈ B h Fn d ∧ ∀ x ∈ d h ⁡ x = G ⁡ x h ↾ pred x A R
13 9 12 eleq2s ⊢ h ∈ C → ∃ d ∈ B h Fn d
14 6 13 bnj771 ⊢ φ → ∃ d ∈ B h Fn d