Theorem bnj1448

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	bnj1448.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
	bnj1448.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1448.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
	bnj1448.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
	bnj1448.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
	bnj1448.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
	bnj1448.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
	bnj1448.8	No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
	bnj1448.9	No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
	bnj1448.10	$$⊢P=\bigcup H$$
	bnj1448.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1448.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
	bnj1448.13	$$⊢W=⟨z,{Q\upharpoonright }_{pred\left(z,A,R\right)}⟩$$
Assertion	bnj1448	$$⊢Q\left(z\right)=G\left(W\right)\to \forall fQ\left(z\right)=G\left(W\right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj1448.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
2		bnj1448.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
3		bnj1448.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
4		bnj1448.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
5		bnj1448.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
6		bnj1448.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
7		bnj1448.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
8		bnj1448.8	Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9		bnj1448.9	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		bnj1448.10	$$⊢P=\bigcup H$$
11		bnj1448.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
12		bnj1448.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
13		bnj1448.13	$$⊢W=⟨z,{Q\upharpoonright }_{pred\left(z,A,R\right)}⟩$$
14	9	bnj1317	$$⊢w\in H\to \forall fw\in H$$
15	14	nfcii	$$⊢\underset{\_}{Ⅎ}fH$$
16	15	nfuni	$$⊢\underset{\_}{Ⅎ}f\bigcup H$$
17	10 16	nfcxfr	$$⊢\underset{\_}{Ⅎ}fP$$
18		nfcv	$$⊢\underset{\_}{Ⅎ}fx$$
19		nfcv	$$⊢\underset{\_}{Ⅎ}fG$$
20		nfcv	$$⊢\underset{\_}{Ⅎ}fpred\left(x,A,R\right)$$
21	17 20	nfres	$$⊢\underset{\_}{Ⅎ}f\left({P\upharpoonright }_{pred\left(x,A,R\right)}\right)$$
22	18 21	nfop	$$⊢\underset{\_}{Ⅎ}f⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
23	11 22	nfcxfr	$$⊢\underset{\_}{Ⅎ}fZ$$
24	19 23	nffv	$$⊢\underset{\_}{Ⅎ}fG\left(Z\right)$$
25	18 24	nfop	$$⊢\underset{\_}{Ⅎ}f⟨x,G\left(Z\right)⟩$$
26	25	nfsn	$$⊢\underset{\_}{Ⅎ}f\left\{⟨x,G\left(Z\right)⟩\right\}$$
27	17 26	nfun	$$⊢\underset{\_}{Ⅎ}f\left(P\cup \left\{⟨x,G\left(Z\right)⟩\right\}\right)$$
28	12 27	nfcxfr	$$⊢\underset{\_}{Ⅎ}fQ$$
29		nfcv	$$⊢\underset{\_}{Ⅎ}fz$$
30	28 29	nffv	$$⊢\underset{\_}{Ⅎ}fQ\left(z\right)$$
31		nfcv	$$⊢\underset{\_}{Ⅎ}fpred\left(z,A,R\right)$$
32	28 31	nfres	$$⊢\underset{\_}{Ⅎ}f\left({Q\upharpoonright }_{pred\left(z,A,R\right)}\right)$$
33	29 32	nfop	$$⊢\underset{\_}{Ⅎ}f⟨z,{Q\upharpoonright }_{pred\left(z,A,R\right)}⟩$$
34	13 33	nfcxfr	$$⊢\underset{\_}{Ⅎ}fW$$
35	19 34	nffv	$$⊢\underset{\_}{Ⅎ}fG\left(W\right)$$
36	30 35	nfeq	$$⊢ℲfQ\left(z\right)=G\left(W\right)$$
37	36	nf5ri	$$⊢Q\left(z\right)=G\left(W\right)\to \forall fQ\left(z\right)=G\left(W\right)$$