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 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
	bnj958.2	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
Assertion	bnj958	⊢ ( ( 𝐺 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑖 ) → ∀ 𝑦 ( 𝐺 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑖 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj958.1	⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
2		bnj958.2	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
3		nfcv	⊢ Ⅎ 𝑦 𝑓
4		nfcv	⊢ Ⅎ 𝑦 𝑛
5		nfiu1	⊢ Ⅎ 𝑦 ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
6	1 5	nfcxfr	⊢ Ⅎ 𝑦 𝐶
7	4 6	nfop	⊢ Ⅎ 𝑦 ⟨ 𝑛 , 𝐶 ⟩
8	7	nfsn	⊢ Ⅎ 𝑦 { ⟨ 𝑛 , 𝐶 ⟩ }
9	3 8	nfun	⊢ Ⅎ 𝑦 ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
10	2 9	nfcxfr	⊢ Ⅎ 𝑦 𝐺
11		nfcv	⊢ Ⅎ 𝑦 𝑖
12	10 11	nffv	⊢ Ⅎ 𝑦 ( 𝐺 ‘ 𝑖 )
13	12	nfeq1	⊢ Ⅎ 𝑦 ( 𝐺 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑖 )
14	13	nf5ri	⊢ ( ( 𝐺 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑖 ) → ∀ 𝑦 ( 𝐺 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑖 ) )