Theorem bnj1416

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	bnj1416.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
	bnj1416.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1416.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
	bnj1416.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
	bnj1416.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
	bnj1416.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
	bnj1416.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
	bnj1416.8	No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
	bnj1416.9	No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
	bnj1416.10	$$⊢P=\bigcup H$$
	bnj1416.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
	bnj1416.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
	bnj1416.28	$$⊢\chi \to \mathrm{dom}P=trCl\left(x,A,R\right)$$
Assertion	bnj1416	$$⊢\chi \to \mathrm{dom}Q=\left\{x\right\}\cup trCl\left(x,A,R\right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj1416.1	$$⊢B=\left\{d|\left(d\subseteq A\wedge \forall x\in dpred\left(x,A,R\right)\subseteq d\right)\right\}$$
2		bnj1416.2	$$⊢Y=⟨x,{f\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
3		bnj1416.3	$$⊢C=\left\{f|\exists d\in B\left(fFnd\wedge \forall x\in df\left(x\right)=G\left(Y\right)\right)\right\}$$
4		bnj1416.4	$$⊢\tau \leftrightarrow \left(f\in C\wedge \mathrm{dom}f=\left\{x\right\}\cup trCl\left(x,A,R\right)\right)$$
5		bnj1416.5	$$⊢D=\left\{x\in A|\neg \exists f\tau \right\}$$
6		bnj1416.6	$$⊢\psi \leftrightarrow \left(RFrSeA\wedge D\ne \varnothing \right)$$
7		bnj1416.7	$$⊢\chi \leftrightarrow \left(\psi \wedge x\in D\wedge \forall y\in D\neg yRx\right)$$
8		bnj1416.8	Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9		bnj1416.9	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		bnj1416.10	$$⊢P=\bigcup H$$
11		bnj1416.11	$$⊢Z=⟨x,{P\upharpoonright }_{pred\left(x,A,R\right)}⟩$$
12		bnj1416.12	$$⊢Q=P\cup \left\{⟨x,G\left(Z\right)⟩\right\}$$
13		bnj1416.28	$$⊢\chi \to \mathrm{dom}P=trCl\left(x,A,R\right)$$
14	12	dmeqi	$$⊢\mathrm{dom}Q=\mathrm{dom}\left(P\cup \left\{⟨x,G\left(Z\right)⟩\right\}\right)$$
15		dmun	$$⊢\mathrm{dom}\left(P\cup \left\{⟨x,G\left(Z\right)⟩\right\}\right)=\mathrm{dom}P\cup \mathrm{dom}\left\{⟨x,G\left(Z\right)⟩\right\}$$
16		fvex	$$⊢G\left(Z\right)\in \mathrm{V}$$
17	16	dmsnop	$$⊢\mathrm{dom}\left\{⟨x,G\left(Z\right)⟩\right\}=\left\{x\right\}$$
18	17	uneq2i	$$⊢\mathrm{dom}P\cup \mathrm{dom}\left\{⟨x,G\left(Z\right)⟩\right\}=\mathrm{dom}P\cup \left\{x\right\}$$
19	14 15 18	3eqtri	$$⊢\mathrm{dom}Q=\mathrm{dom}P\cup \left\{x\right\}$$
20	13	uneq1d	$$⊢\chi \to \mathrm{dom}P\cup \left\{x\right\}=trCl\left(x,A,R\right)\cup \left\{x\right\}$$
21		uncom	$$⊢trCl\left(x,A,R\right)\cup \left\{x\right\}=\left\{x\right\}\cup trCl\left(x,A,R\right)$$
22	20 21	eqtrdi	$$⊢\chi \to \mathrm{dom}P\cup \left\{x\right\}=\left\{x\right\}\cup trCl\left(x,A,R\right)$$
23	19 22	syl5eq	$$⊢\chi \to \mathrm{dom}Q=\left\{x\right\}\cup trCl\left(x,A,R\right)$$