Theorem nfcoll

Description: Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023)

	Ref	Expression
Hypotheses	nfcoll.1	⊢ Ⅎ 𝑥 𝐹
	nfcoll.2	⊢ Ⅎ 𝑥 𝐴
Assertion	nfcoll	⊢ Ⅎ 𝑥 ( 𝐹 Coll 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nfcoll.1	⊢ Ⅎ 𝑥 𝐹
2		nfcoll.2	⊢ Ⅎ 𝑥 𝐴
3		df-coll	⊢ ( 𝐹 Coll 𝐴 ) = ∪ 𝑦 ∈ 𝐴 Scott ( 𝐹 “ { 𝑦 } )
4		nfcv	⊢ Ⅎ 𝑥 { 𝑦 }
5	1 4	nfima	⊢ Ⅎ 𝑥 ( 𝐹 “ { 𝑦 } )
6	5	nfscott	⊢ Ⅎ 𝑥 Scott ( 𝐹 “ { 𝑦 } )
7	2 6	nfiun	⊢ Ⅎ 𝑥 ∪ 𝑦 ∈ 𝐴 Scott ( 𝐹 “ { 𝑦 } )
8	3 7	nfcxfr	⊢ Ⅎ 𝑥 ( 𝐹 Coll 𝐴 )