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Metamath Proof Explorer

Theorem nffv

Description: Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995) (Revised by Mario Carneiro, 15-Oct-2016)

Ref Expression
Hypotheses nffv.1 ⊢ Ⅎ 𝑥 𝐹
nffv.2 ⊢ Ⅎ 𝑥 𝐴
Assertion nffv Ⅎ 𝑥 ( 𝐹 ‘ 𝐴 )

Proof

Step Hyp Ref Expression
1 nffv.1 ⊢ Ⅎ 𝑥 𝐹
2 nffv.2 ⊢ Ⅎ 𝑥 𝐴
3 df-fv ⊢ ( 𝐹 ‘ 𝐴 ) = ( ℩ 𝑦 𝐴 𝐹 𝑦 )
4 nfcv ⊢ Ⅎ 𝑥 𝑦
5 2 1 4 nfbr ⊢ Ⅎ 𝑥 𝐴 𝐹 𝑦
6 5 nfiotaw ⊢ Ⅎ 𝑥 ( ℩ 𝑦 𝐴 𝐹 𝑦 )
7 3 6 nfcxfr ⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝐴 )