bnj130

Description: Technical lemma for bnj151 . 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	bnj130.1	$⊢ θ \leftrightarrow ((R FrSe A \land x \in A) \to \exists! f (f Fn n \land φ \land ψ))$
	bnj130.2	No typesetting found for \|- ( ph' <-> [. 1o / n ]. ph ) with typecode \|-
	bnj130.3	No typesetting found for \|- ( ps' <-> [. 1o / n ]. ps ) with typecode \|-
	bnj130.4	No typesetting found for \|- ( th' <-> [. 1o / n ]. th ) with typecode \|-
Assertion	bnj130	Could not format assertion : No typesetting found for \|- ( th' <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn 1o /\ ph' /\ ps' ) ) ) with typecode \|-

Proof

Step	Hyp	Ref	Expression
1		bnj130.1	$⊢ θ \leftrightarrow ((R FrSe A \land x \in A) \to \exists! f (f Fn n \land φ \land ψ))$
2		bnj130.2	Could not format ( ph' <-> [. 1o / n ]. ph ) : No typesetting found for \|- ( ph' <-> [. 1o / n ]. ph ) with typecode \|-
3		bnj130.3	Could not format ( ps' <-> [. 1o / n ]. ps ) : No typesetting found for \|- ( ps' <-> [. 1o / n ]. ps ) with typecode \|-
4		bnj130.4	Could not format ( th' <-> [. 1o / n ]. th ) : No typesetting found for \|- ( th' <-> [. 1o / n ]. th ) with typecode \|-
5	1	sbcbii	$⊢ \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} θ \leftrightarrow \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} ((R FrSe A \land x \in A) \to \exists! f (f Fn n \land φ \land ψ))$
6		bnj105	$⊢ 1_{𝑜} \in V$
7	6	bnj90	$⊢ \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} f Fn n \leftrightarrow f Fn 1_{𝑜}$
8	7	bicomi	$⊢ f Fn 1_{𝑜} \leftrightarrow \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} f Fn n$
9	8 2 3	3anbi123i	Could not format ( ( f Fn 1o /\ ph' /\ ps' ) <-> ( [. 1o / n ]. f Fn n /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) ) : No typesetting found for \|- ( ( f Fn 1o /\ ph' /\ ps' ) <-> ( [. 1o / n ]. f Fn n /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) ) with typecode \|-
10		sbc3an	$⊢ \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} (f Fn n \land φ \land ψ) \leftrightarrow (\underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} f Fn n \land \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} φ \land \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} ψ)$
11	9 10	bitr4i	Could not format ( ( f Fn 1o /\ ph' /\ ps' ) <-> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) : No typesetting found for \|- ( ( f Fn 1o /\ ph' /\ ps' ) <-> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) with typecode \|-
12	11	eubii	Could not format ( E! f ( f Fn 1o /\ ph' /\ ps' ) <-> E! f [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) : No typesetting found for \|- ( E! f ( f Fn 1o /\ ph' /\ ps' ) <-> E! f [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) with typecode \|-
13	6	bnj89	$⊢ \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} \exists! f (f Fn n \land φ \land ψ) \leftrightarrow \exists! f \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} (f Fn n \land φ \land ψ)$
14	12 13	bitr4i	Could not format ( E! f ( f Fn 1o /\ ph' /\ ps' ) <-> [. 1o / n ]. E! f ( f Fn n /\ ph /\ ps ) ) : No typesetting found for \|- ( E! f ( f Fn 1o /\ ph' /\ ps' ) <-> [. 1o / n ]. E! f ( f Fn n /\ ph /\ ps ) ) with typecode \|-
15	14	imbi2i	Could not format ( ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn 1o /\ ph' /\ ps' ) ) <-> ( ( R _FrSe A /\ x e. A ) -> [. 1o / n ]. E! f ( f Fn n /\ ph /\ ps ) ) ) : No typesetting found for \|- ( ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn 1o /\ ph' /\ ps' ) ) <-> ( ( R _FrSe A /\ x e. A ) -> [. 1o / n ]. E! f ( f Fn n /\ ph /\ ps ) ) ) with typecode \|-
16		nfv	$⊢ Ⅎ n (R FrSe A \land x \in A)$
17	16	sbc19.21g	$⊢ 1_{𝑜} \in V \to (\underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} ((R FrSe A \land x \in A) \to \exists! f (f Fn n \land φ \land ψ)) \leftrightarrow ((R FrSe A \land x \in A) \to \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} \exists! f (f Fn n \land φ \land ψ)))$
18	6 17	ax-mp	$⊢ \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} ((R FrSe A \land x \in A) \to \exists! f (f Fn n \land φ \land ψ)) \leftrightarrow ((R FrSe A \land x \in A) \to \underset{˙}{[} 1_{𝑜} / n \underset{˙}{]} \exists! f (f Fn n \land φ \land ψ))$
19	15 18	bitr4i	Could not format ( ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn 1o /\ ph' /\ ps' ) ) <-> [. 1o / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) ) : No typesetting found for \|- ( ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn 1o /\ ph' /\ ps' ) ) <-> [. 1o / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) ) with typecode \|-
20	5 4 19	3bitr4i	Could not format ( th' <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn 1o /\ ph' /\ ps' ) ) ) : No typesetting found for \|- ( th' <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn 1o /\ ph' /\ ps' ) ) ) with typecode \|-

Metamath Proof Explorer

Theorem bnj130

Proof