Theorem 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	$$⊢\theta \leftrightarrow \left(\left(RFrSeA\wedge x\in A\right)\to \exists !f\left(fFnn\wedge \phi \wedge \psi \right)\right)$$
	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	$$⊢\theta \leftrightarrow \left(\left(RFrSeA\wedge x\in A\right)\to \exists !f\left(fFnn\wedge \phi \wedge \psi \right)\right)$$
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	$$⊢\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\theta \leftrightarrow \underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\left(\left(RFrSeA\wedge x\in A\right)\to \exists !f\left(fFnn\wedge \phi \wedge \psi \right)\right)$$
6		bnj105	$$⊢1_{𝑜}\in \mathrm{V}$$
7	6	bnj90	$$⊢\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}fFnn\leftrightarrow fFn{1}_{𝑜}$$
8	7	bicomi	$$⊢fFn{1}_{𝑜}\leftrightarrow \underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}fFnn$$
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	$$⊢\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\left(fFnn\wedge \phi \wedge \psi \right)\leftrightarrow \left(\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}fFnn\wedge \underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\phi \wedge \underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\psi \right)$$
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	$$⊢\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\exists !f\left(fFnn\wedge \phi \wedge \psi \right)\leftrightarrow \exists !f\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\left(fFnn\wedge \phi \wedge \psi \right)$$
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	$$⊢Ⅎn\left(RFrSeA\wedge x\in A\right)$$
17	16	sbc19.21g	$$⊢1_{𝑜}\in \mathrm{V}\to \left(\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\left(\left(RFrSeA\wedge x\in A\right)\to \exists !f\left(fFnn\wedge \phi \wedge \psi \right)\right)\leftrightarrow \left(\left(RFrSeA\wedge x\in A\right)\to \underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\exists !f\left(fFnn\wedge \phi \wedge \psi \right)\right)\right)$$
18	6 17	ax-mp	$$⊢\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\left(\left(RFrSeA\wedge x\in A\right)\to \exists !f\left(fFnn\wedge \phi \wedge \psi \right)\right)\leftrightarrow \left(\left(RFrSeA\wedge x\in A\right)\to \underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\exists !f\left(fFnn\wedge \phi \wedge \psi \right)\right)$$
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 |-