Theorem bnj958

Description: Technical lemma for bnj69 . 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	bnj958.1	$$⊢C=\bigcup _{y\in f\left(m\right)}pred\left(y,A,R\right)$$
	bnj958.2	$$⊢G=f\cup \left\{⟨n,C⟩\right\}$$
Assertion	bnj958	$$⊢G\left(i\right)=f\left(i\right)\to \forall yG\left(i\right)=f\left(i\right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj958.1	$$⊢C=\bigcup _{y\in f\left(m\right)}pred\left(y,A,R\right)$$
2		bnj958.2	$$⊢G=f\cup \left\{⟨n,C⟩\right\}$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}yf$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}yn$$
5		nfiu1	$$⊢\underset{\_}{Ⅎ}y\bigcup _{y\in f\left(m\right)}pred\left(y,A,R\right)$$
6	1 5	nfcxfr	$$⊢\underset{\_}{Ⅎ}yC$$
7	4 6	nfop	$$⊢\underset{\_}{Ⅎ}y⟨n,C⟩$$
8	7	nfsn	$$⊢\underset{\_}{Ⅎ}y\left\{⟨n,C⟩\right\}$$
9	3 8	nfun	$$⊢\underset{\_}{Ⅎ}y\left(f\cup \left\{⟨n,C⟩\right\}\right)$$
10	2 9	nfcxfr	$$⊢\underset{\_}{Ⅎ}yG$$
11		nfcv	$$⊢\underset{\_}{Ⅎ}yi$$
12	10 11	nffv	$$⊢\underset{\_}{Ⅎ}yG\left(i\right)$$
13	12	nfeq1	$$⊢ℲyG\left(i\right)=f\left(i\right)$$
14	13	nf5ri	$$⊢G\left(i\right)=f\left(i\right)\to \forall yG\left(i\right)=f\left(i\right)$$