Theorem bnj1449

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	bnj1449.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
	bnj1449.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1449.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
	bnj1449.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
	bnj1449.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
	bnj1449.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
	bnj1449.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
	bnj1449.8	No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
	bnj1449.9	No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
	bnj1449.10	$$⊢P=\bigcup H$$
	bnj1449.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1449.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
	bnj1449.13	$$⊢W=⟨z,{Q\upharpoonright }_{pred\left(z,A,R\right)}⟩$$
	bnj1449.14	$$⊢E=\left\{x\right\}\cup trCl\left(x,A,R\right)$$
	bnj1449.15	$$⊢\chi \to PFntrCl\left(x,A,R\right)$$
	bnj1449.16	$$⊢\chi \to QFn\left(\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
	bnj1449.17	$$⊢\theta \leftrightarrow \left(\chi \wedge z\in E\right)$$
	bnj1449.18	$$⊢\eta \leftrightarrow \left(\theta \wedge z\in \left\{x\right\}\right)$$
	bnj1449.19	$$⊢\zeta \leftrightarrow \left(\theta \wedge z\in trCl\left(x,A,R\right)\right)$$
Assertion	bnj1449	$$⊢\zeta \to \forall f\zeta$$

Proof

Step	Hyp	Ref	Expression
1		bnj1449.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
2		bnj1449.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
3		bnj1449.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
4		bnj1449.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
5		bnj1449.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
6		bnj1449.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
7		bnj1449.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
8		bnj1449.8	Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9		bnj1449.9	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		bnj1449.10	$$⊢P=\bigcup H$$
11		bnj1449.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
12		bnj1449.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
13		bnj1449.13	$$⊢W=⟨z,{Q\upharpoonright }_{pred\left(z,A,R\right)}⟩$$
14		bnj1449.14	$$⊢E=\left\{x\right\}\cup trCl\left(x,A,R\right)$$
15		bnj1449.15	$$⊢\chi \to PFntrCl\left(x,A,R\right)$$
16		bnj1449.16	$$⊢\chi \to QFn\left(\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
17		bnj1449.17	$$⊢\theta \leftrightarrow \left(\chi \wedge z\in E\right)$$
18		bnj1449.18	$$⊢\eta \leftrightarrow \left(\theta \wedge z\in \left\{x\right\}\right)$$
19		bnj1449.19	$$⊢\zeta \leftrightarrow \left(\theta \wedge z\in trCl\left(x,A,R\right)\right)$$
20		nfv	$$⊢ℲfRFrSeA$$
21		nfe1	$$⊢Ⅎf\exists f\tau$$
22	21	nfn	$$⊢Ⅎf\neg \exists f\tau$$
23		nfcv	$$⊢\underset{\_}{Ⅎ}fA$$
24	22 23	nfrabw	$$⊢\underset{\_}{Ⅎ}f\left\{x\in A|\neg \exists f\tau \right\}$$
25	5 24	nfcxfr	$$⊢\underset{\_}{Ⅎ}fD$$
26		nfcv	$$⊢\underset{\_}{Ⅎ}f\varnothing$$
27	25 26	nfne	$$⊢ℲfD\ne \varnothing$$
28	20 27	nfan	$$⊢Ⅎf\left(RFrSeA\wedge D\ne \varnothing \right)$$
29	6 28	nfxfr	$$⊢Ⅎf\psi$$
30	25	nfcri	$$⊢Ⅎfx\in D$$
31		nfv	$$⊢Ⅎf\neg yRx$$
32	25 31	nfralw	$$⊢Ⅎf\forall y\in D\neg yRx$$
33	29 30 32	nf3an	$$⊢Ⅎf\left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
34	7 33	nfxfr	$$⊢Ⅎf\chi$$
35		nfv	$$⊢Ⅎfz\in E$$
36	34 35	nfan	$$⊢Ⅎf\left(\chi \wedge z\in E\right)$$
37	17 36	nfxfr	$$⊢Ⅎf\theta$$
38		nfv	$$⊢Ⅎfz\in trCl\left(x,A,R\right)$$
39	37 38	nfan	$$⊢Ⅎf\left(\theta \wedge z\in trCl\left(x,A,R\right)\right)$$
40	19 39	nfxfr	$$⊢Ⅎf\zeta$$
41	40	nf5ri	$$⊢\zeta \to \forall f\zeta$$