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Metamath Proof Explorer

Theorem nfse

Description: Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015) (Revised by Mario Carneiro, 14-Oct-2016)

Ref Expression
Hypotheses nffr.r ⊢ Ⅎ 𝑥 𝑅
nffr.a ⊢ Ⅎ 𝑥 𝐴
Assertion nfse Ⅎ 𝑥 𝑅 Se 𝐴

Proof

Step Hyp Ref Expression
1 nffr.r ⊢ Ⅎ 𝑥 𝑅
2 nffr.a ⊢ Ⅎ 𝑥 𝐴
3 df-se ⊢ ( 𝑅 Se 𝐴 ↔ ∀ 𝑏 ∈ 𝐴 { 𝑎 ∈ 𝐴 ∣ 𝑎 𝑅 𝑏 } ∈ V )
4 nfcv ⊢ Ⅎ 𝑥 𝑎
5 nfcv ⊢ Ⅎ 𝑥 𝑏
6 4 1 5 nfbr ⊢ Ⅎ 𝑥 𝑎 𝑅 𝑏
7 6 2 nfrabw ⊢ Ⅎ 𝑥 { 𝑎 ∈ 𝐴 ∣ 𝑎 𝑅 𝑏 }
8 7 nfel1 ⊢ Ⅎ 𝑥 { 𝑎 ∈ 𝐴 ∣ 𝑎 𝑅 𝑏 } ∈ V
9 2 8 nfralw ⊢ Ⅎ 𝑥 ∀ 𝑏 ∈ 𝐴 { 𝑎 ∈ 𝐴 ∣ 𝑎 𝑅 𝑏 } ∈ V
10 3 9 nfxfr ⊢ Ⅎ 𝑥 𝑅 Se 𝐴