Theorem nfafv2

Description: Bound-variable hypothesis builder for function value, analogous to nffv . To prove a deduction version of this analogous to nffvd is not easily possible because a deduction version of nfdfat cannot be shown easily. (Contributed by AV, 4-Sep-2022)

	Ref	Expression
Hypotheses	nfafv2.1	⊢ Ⅎ 𝑥 𝐹
	nfafv2.2	⊢ Ⅎ 𝑥 𝐴
Assertion	nfafv2	⊢ Ⅎ 𝑥 ( 𝐹 '''' 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nfafv2.1	⊢ Ⅎ 𝑥 𝐹
2		nfafv2.2	⊢ Ⅎ 𝑥 𝐴
3		df-afv2	⊢ ( 𝐹 '''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( ℩ 𝑦 𝐴 𝐹 𝑦 ) , 𝒫 ∪ ran 𝐹 )
4	1 2	nfdfat	⊢ Ⅎ 𝑥 𝐹 defAt 𝐴
5		nfcv	⊢ Ⅎ 𝑥 𝑦
6	2 1 5	nfbr	⊢ Ⅎ 𝑥 𝐴 𝐹 𝑦
7	6	nfiotaw	⊢ Ⅎ 𝑥 ( ℩ 𝑦 𝐴 𝐹 𝑦 )
8	1	nfrn	⊢ Ⅎ 𝑥 ran 𝐹
9	8	nfuni	⊢ Ⅎ 𝑥 ∪ ran 𝐹
10	9	nfpw	⊢ Ⅎ 𝑥 𝒫 ∪ ran 𝐹
11	4 7 10	nfif	⊢ Ⅎ 𝑥 if ( 𝐹 defAt 𝐴 , ( ℩ 𝑦 𝐴 𝐹 𝑦 ) , 𝒫 ∪ ran 𝐹 )
12	3 11	nfcxfr	⊢ Ⅎ 𝑥 ( 𝐹 '''' 𝐴 )