Theorem bnj1466

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	bnj1466.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
	bnj1466.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1466.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
	bnj1466.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
	bnj1466.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
	bnj1466.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
	bnj1466.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
	bnj1466.8	No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
	bnj1466.9	No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
	bnj1466.10	$$⊢P=\bigcup H$$
	bnj1466.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1466.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
Assertion	bnj1466	$$⊢w\in Q\to \forall fw\in Q$$

Proof

Step	Hyp	Ref	Expression
1		bnj1466.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
2		bnj1466.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
3		bnj1466.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
4		bnj1466.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
5		bnj1466.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
6		bnj1466.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
7		bnj1466.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
8		bnj1466.8	Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9		bnj1466.9	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		bnj1466.10	$$⊢P=\bigcup H$$
11		bnj1466.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
12		bnj1466.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
13	9	bnj1317	$$⊢w\in H\to \forall fw\in H$$
14	13	nfcii	$$⊢\underset{\_}{Ⅎ}fH$$
15	14	nfuni	$$⊢\underset{\_}{Ⅎ}f\bigcup H$$
16	10 15	nfcxfr	$$⊢\underset{\_}{Ⅎ}fP$$
17		nfcv	$$⊢\underset{\_}{Ⅎ}fx$$
18		nfcv	$$⊢\underset{\_}{Ⅎ}fG$$
19		nfcv	$$⊢\underset{\_}{Ⅎ}fpred\left(x,A,R\right)$$
20	16 19	nfres	$$⊢\underset{\_}{Ⅎ}f\left({P\upharpoonright }_{pred\left(x,A,R\right)}\right)$$
21	17 20	nfop	$$⊢\underset{\_}{Ⅎ}f⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
22	11 21	nfcxfr	$$⊢\underset{\_}{Ⅎ}fZ$$
23	18 22	nffv	$$⊢\underset{\_}{Ⅎ}fG\left(Z\right)$$
24	17 23	nfop	$$⊢\underset{\_}{Ⅎ}f⟨x,G\left(Z\right)⟩$$
25	24	nfsn	$$⊢\underset{\_}{Ⅎ}f\left\{⟨x,G\left(Z\right)⟩\right\}$$
26	16 25	nfun	$$⊢\underset{\_}{Ⅎ}f\left(P\cup \left\{⟨x,G\left(Z\right)⟩\right\}\right)$$
27	12 26	nfcxfr	$$⊢\underset{\_}{Ⅎ}fQ$$
28	27	nfcrii	$$⊢w\in Q\to \forall fw\in Q$$