Theorem bnj1446

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	bnj1446.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
	bnj1446.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1446.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
	bnj1446.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
	bnj1446.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
	bnj1446.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
	bnj1446.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
	bnj1446.8	No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
	bnj1446.9	No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
	bnj1446.10	$$⊢P=\bigcup H$$
	bnj1446.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1446.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
	bnj1446.13	$$⊢W=⟨z,{Q\upharpoonright }_{pred\left(z,A,R\right)}⟩$$
Assertion	bnj1446	$$⊢Q\left(z\right)=G\left(W\right)\to \forall dQ\left(z\right)=G\left(W\right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj1446.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
2		bnj1446.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
3		bnj1446.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
4		bnj1446.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
5		bnj1446.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
6		bnj1446.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
7		bnj1446.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
8		bnj1446.8	Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9		bnj1446.9	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		bnj1446.10	$$⊢P=\bigcup H$$
11		bnj1446.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
12		bnj1446.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
13		bnj1446.13	$$⊢W=⟨z,{Q\upharpoonright }_{pred\left(z,A,R\right)}⟩$$
14		nfcv	$$⊢\underset{\_}{Ⅎ}dpred\left(x,A,R\right)$$
15		nfcv	$$⊢\underset{\_}{Ⅎ}dy$$
16		nfre1	$$⊢Ⅎd\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)$$
17	16	nfab	$$⊢\underset{\_}{Ⅎ}d\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
18	3 17	nfcxfr	$$⊢\underset{\_}{Ⅎ}dC$$
19	18	nfcri	$$⊢Ⅎdf\in C$$
20		nfv	$$⊢Ⅎd\mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)$$
21	19 20	nfan	$$⊢Ⅎd\left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
22	4 21	nfxfr	$$⊢Ⅎd\tau$$
23	15 22	nfsbcw	$$⊢Ⅎd\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\tau$$
24	8 23	nfxfr	Could not format F/ d ta' : No typesetting found for |- F/ d ta' with typecode |-
25	14 24	nfrexw	Could not format F/ d E. y e. _pred ( x , A , R ) ta' : No typesetting found for |- F/ d E. y e. _pred ( x , A , R ) ta' with typecode |-
26	25	nfab	Could not format F/_ d { f | E. y e. _pred ( x , A , R ) ta' } : No typesetting found for |- F/_ d { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
27	9 26	nfcxfr	$$⊢\underset{\_}{Ⅎ}dH$$
28	27	nfuni	$$⊢\underset{\_}{Ⅎ}d\bigcup H$$
29	10 28	nfcxfr	$$⊢\underset{\_}{Ⅎ}dP$$
30		nfcv	$$⊢\underset{\_}{Ⅎ}dx$$
31		nfcv	$$⊢\underset{\_}{Ⅎ}dG$$
32	29 14	nfres	$$⊢\underset{\_}{Ⅎ}d\left({P\upharpoonright }_{pred\left(x,A,R\right)}\right)$$
33	30 32	nfop	$$⊢\underset{\_}{Ⅎ}d⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
34	11 33	nfcxfr	$$⊢\underset{\_}{Ⅎ}dZ$$
35	31 34	nffv	$$⊢\underset{\_}{Ⅎ}dG\left(Z\right)$$
36	30 35	nfop	$$⊢\underset{\_}{Ⅎ}d⟨x,G\left(Z\right)⟩$$
37	36	nfsn	$$⊢\underset{\_}{Ⅎ}d\left\{⟨x,G\left(Z\right)⟩\right\}$$
38	29 37	nfun	$$⊢\underset{\_}{Ⅎ}d\left(P\cup \left\{⟨x,G\left(Z\right)⟩\right\}\right)$$
39	12 38	nfcxfr	$$⊢\underset{\_}{Ⅎ}dQ$$
40		nfcv	$$⊢\underset{\_}{Ⅎ}dz$$
41	39 40	nffv	$$⊢\underset{\_}{Ⅎ}dQ\left(z\right)$$
42		nfcv	$$⊢\underset{\_}{Ⅎ}dpred\left(z,A,R\right)$$
43	39 42	nfres	$$⊢\underset{\_}{Ⅎ}d\left({Q\upharpoonright }_{pred\left(z,A,R\right)}\right)$$
44	40 43	nfop	$$⊢\underset{\_}{Ⅎ}d⟨z,{Q\upharpoonright }_{pred\left(z,A,R\right)}⟩$$
45	13 44	nfcxfr	$$⊢\underset{\_}{Ⅎ}dW$$
46	31 45	nffv	$$⊢\underset{\_}{Ⅎ}dG\left(W\right)$$
47	41 46	nfeq	$$⊢ℲdQ\left(z\right)=G\left(W\right)$$
48	47	nf5ri	$$⊢Q\left(z\right)=G\left(W\right)\to \forall dQ\left(z\right)=G\left(W\right)$$