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Metamath Proof Explorer

Theorem bnj1256

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 bnj1256.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
bnj1256.2 ⊢ Y = x f ↾ pred x A R
bnj1256.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
bnj1256.4 ⊢ D = dom ⁡ g ∩ dom ⁡ h
bnj1256.5 ⊢ E = x ∈ D | g ⁡ x ≠ h ⁡ x
bnj1256.6 ⊢ φ ↔ R FrSe A ∧ g ∈ C ∧ h ∈ C ∧ g ↾ D ≠ h ↾ D
bnj1256.7 ⊢ ψ ↔ φ ∧ x ∈ E ∧ ∀ y ∈ E ¬ y R x
Assertion bnj1256 ⊢ φ → ∃ d ∈ B g Fn d

Proof

Step Hyp Ref Expression
1 bnj1256.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
2 bnj1256.2 ⊢ Y = x f ↾ pred x A R
3 bnj1256.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
4 bnj1256.4 ⊢ D = dom ⁡ g ∩ dom ⁡ h
5 bnj1256.5 ⊢ E = x ∈ D | g ⁡ x ≠ h ⁡ x
6 bnj1256.6 ⊢ φ ↔ R FrSe A ∧ g ∈ C ∧ h ∈ C ∧ g ↾ D ≠ h ↾ D
7 bnj1256.7 ⊢ ψ ↔ φ ∧ x ∈ E ∧ ∀ y ∈ E ¬ y R x
8 abid ⊢ g ∈ g | ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ x g ↾ pred x A R ↔ ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ x g ↾ pred x A R
9 8 bnj1238 ⊢ g ∈ g | ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ x g ↾ pred x A R → ∃ d ∈ B g Fn d
10 eqid ⊢ x g ↾ pred x A R = x g ↾ pred x A R
11 eqid ⊢ g | ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ x g ↾ pred x A R = g | ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ x g ↾ pred x A R
12 2 3 10 11 bnj1234 ⊢ C = g | ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ x g ↾ pred x A R
13 9 12 eleq2s ⊢ g ∈ C → ∃ d ∈ B g Fn d
14 6 13 bnj770 ⊢ φ → ∃ d ∈ B g Fn d