Theorem bnj1374

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	bnj1374.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
	bnj1374.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1374.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
	bnj1374.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
	bnj1374.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
	bnj1374.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
	bnj1374.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
	bnj1374.8	No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
	bnj1374.9	No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
Assertion	bnj1374	$$⊢f\in H\to f\in C$$

Proof

Step	Hyp	Ref	Expression
1		bnj1374.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
2		bnj1374.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
3		bnj1374.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
4		bnj1374.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
5		bnj1374.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
6		bnj1374.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
7		bnj1374.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
8		bnj1374.8	Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9		bnj1374.9	Could not format H = { f | E. y e. _pred ( x , A , R ) ta' } : No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
10	9	bnj1436	Could not format ( f e. H -> E. y e. _pred ( x , A , R ) ta' ) : No typesetting found for |- ( f e. H -> E. y e. _pred ( x , A , R ) ta' ) with typecode |-
11		rexex	Could not format ( E. y e. _pred ( x , A , R ) ta' -> E. y ta' ) : No typesetting found for |- ( E. y e. _pred ( x , A , R ) ta' -> E. y ta' ) with typecode |-
12	10 11	syl	Could not format ( f e. H -> E. y ta' ) : No typesetting found for |- ( f e. H -> E. y ta' ) with typecode |-
13	1 2 3 4 8	bnj1373	Could not format ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) : No typesetting found for |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) with typecode |-
14	13	exbii	Could not format ( E. y ta' <-> E. y ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) : No typesetting found for |- ( E. y ta' <-> E. y ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) with typecode |-
15	12 14	sylib	$$⊢f\in H\to \exists y\left(f\in C\wedge \mathrm{dom}f=\left\{y\right\}\cup trCl\left(y,A,R\right)\right)$$
16		exsimpl	$$⊢\exists y\left(f\in C\wedge \mathrm{dom}f=\left\{y\right\}\cup trCl\left(y,A,R\right)\right)\to \exists yf\in C$$
17	15 16	syl	$$⊢f\in H\to \exists yf\in C$$
18	17	bnj937	$$⊢f\in H\to f\in C$$