bnj917

Description: Technical lemma for bnj69 . 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	bnj917.1	$⊢ φ \leftrightarrow f (\emptyset) = pred (X, A, R)$
	bnj917.2	$⊢ ψ \leftrightarrow \forall i \in ω (suc i \in n \to f (suc i) = ⋃_{y \in f (i)} pred (y, A, R))$
	bnj917.3	$⊢ D = ω ∖ \{\emptyset\}$
	bnj917.4	$⊢ B = \{f \| \exists n \in D (f Fn n \land φ \land ψ)\}$
	bnj917.5	$⊢ χ \leftrightarrow (n \in D \land f Fn n \land φ \land ψ)$
Assertion	bnj917	$⊢ y \in trCl (X, A, R) \to \exists f \exists n \exists i (χ \land i \in n \land y \in f (i))$

Proof

Step	Hyp	Ref	Expression
1		bnj917.1	$⊢ φ \leftrightarrow f (\emptyset) = pred (X, A, R)$
2		bnj917.2	$⊢ ψ \leftrightarrow \forall i \in ω (suc i \in n \to f (suc i) = ⋃_{y \in f (i)} pred (y, A, R))$
3		bnj917.3	$⊢ D = ω ∖ \{\emptyset\}$
4		bnj917.4	$⊢ B = \{f \| \exists n \in D (f Fn n \land φ \land ψ)\}$
5		bnj917.5	$⊢ χ \leftrightarrow (n \in D \land f Fn n \land φ \land ψ)$
6		biid	$⊢ (f Fn n \land φ \land ψ) \leftrightarrow (f Fn n \land φ \land ψ)$
7	1 2 3 4 6	bnj916	$⊢ y \in trCl (X, A, R) \to \exists f \exists n \exists i (n \in D \land (f Fn n \land φ \land ψ) \land i \in n \land y \in f (i))$
8		bnj252	$⊢ (n \in D \land f Fn n \land φ \land ψ) \leftrightarrow (n \in D \land (f Fn n \land φ \land ψ))$
9	5 8	bitri	$⊢ χ \leftrightarrow (n \in D \land (f Fn n \land φ \land ψ))$
10	9	3anbi1i	$⊢ (χ \land i \in n \land y \in f (i)) \leftrightarrow ((n \in D \land (f Fn n \land φ \land ψ)) \land i \in n \land y \in f (i))$
11		bnj253	$⊢ (n \in D \land (f Fn n \land φ \land ψ) \land i \in n \land y \in f (i)) \leftrightarrow ((n \in D \land (f Fn n \land φ \land ψ)) \land i \in n \land y \in f (i))$
12	10 11	bitr4i	$⊢ (χ \land i \in n \land y \in f (i)) \leftrightarrow (n \in D \land (f Fn n \land φ \land ψ) \land i \in n \land y \in f (i))$
13	12	3exbii	$⊢ \exists f \exists n \exists i (χ \land i \in n \land y \in f (i)) \leftrightarrow \exists f \exists n \exists i (n \in D \land (f Fn n \land φ \land ψ) \land i \in n \land y \in f (i))$
14	7 13	sylibr	$⊢ y \in trCl (X, A, R) \to \exists f \exists n \exists i (χ \land i \in n \land y \in f (i))$

Metamath Proof Explorer

Theorem bnj917

Proof