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Metamath Proof Explorer

Theorem nfixp

Description: Bound-variable hypothesis builder for indexed Cartesian product. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfixpw when possible. (Contributed by Mario Carneiro, 15-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses nfixp.1 ⊢ Ⅎ 𝑦 𝐴
nfixp.2 ⊢ Ⅎ 𝑦 𝐵
Assertion nfixp Ⅎ 𝑦 X 𝑥 ∈ 𝐴 𝐵

Proof

Step Hyp Ref Expression
1 nfixp.1 ⊢ Ⅎ 𝑦 𝐴
2 nfixp.2 ⊢ Ⅎ 𝑦 𝐵
3 df-ixp ⊢ X 𝑥 ∈ 𝐴 𝐵 = { 𝑧 ∣ ( 𝑧 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑧 ‘ 𝑥 ) ∈ 𝐵 ) }
4 nfcv ⊢ Ⅎ 𝑦 𝑧
5 nftru ⊢ Ⅎ 𝑥 ⊤
6 nfcvf ⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → Ⅎ 𝑦 𝑥 )
7 6 adantl ⊢ ( ( ⊤ ∧ ¬ ∀ 𝑦 𝑦 = 𝑥 ) → Ⅎ 𝑦 𝑥 )
8 1 a1i ⊢ ( ( ⊤ ∧ ¬ ∀ 𝑦 𝑦 = 𝑥 ) → Ⅎ 𝑦 𝐴 )
9 7 8 nfeld ⊢ ( ( ⊤ ∧ ¬ ∀ 𝑦 𝑦 = 𝑥 ) → Ⅎ 𝑦 𝑥 ∈ 𝐴 )
10 5 9 nfabd2 ⊢ ( ⊤ → Ⅎ 𝑦 { 𝑥 ∣ 𝑥 ∈ 𝐴 } )
11 10 mptru ⊢ Ⅎ 𝑦 { 𝑥 ∣ 𝑥 ∈ 𝐴 }
12 4 11 nffn ⊢ Ⅎ 𝑦 𝑧 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 }
13 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑧 ‘ 𝑥 ) ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑧 ‘ 𝑥 ) ∈ 𝐵 ) )
14 4 a1i ⊢ ( ( ⊤ ∧ ¬ ∀ 𝑦 𝑦 = 𝑥 ) → Ⅎ 𝑦 𝑧 )
15 14 7 nffvd ⊢ ( ( ⊤ ∧ ¬ ∀ 𝑦 𝑦 = 𝑥 ) → Ⅎ 𝑦 ( 𝑧 ‘ 𝑥 ) )
16 2 a1i ⊢ ( ( ⊤ ∧ ¬ ∀ 𝑦 𝑦 = 𝑥 ) → Ⅎ 𝑦 𝐵 )
17 15 16 nfeld ⊢ ( ( ⊤ ∧ ¬ ∀ 𝑦 𝑦 = 𝑥 ) → Ⅎ 𝑦 ( 𝑧 ‘ 𝑥 ) ∈ 𝐵 )
18 9 17 nfimd ⊢ ( ( ⊤ ∧ ¬ ∀ 𝑦 𝑦 = 𝑥 ) → Ⅎ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑧 ‘ 𝑥 ) ∈ 𝐵 ) )
19 5 18 nfald2 ⊢ ( ⊤ → Ⅎ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑧 ‘ 𝑥 ) ∈ 𝐵 ) )
20 19 mptru ⊢ Ⅎ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑧 ‘ 𝑥 ) ∈ 𝐵 )
21 13 20 nfxfr ⊢ Ⅎ 𝑦 ∀ 𝑥 ∈ 𝐴 ( 𝑧 ‘ 𝑥 ) ∈ 𝐵
22 12 21 nfan ⊢ Ⅎ 𝑦 ( 𝑧 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑧 ‘ 𝑥 ) ∈ 𝐵 )
23 22 nfab ⊢ Ⅎ 𝑦 { 𝑧 ∣ ( 𝑧 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑧 ‘ 𝑥 ) ∈ 𝐵 ) }
24 3 23 nfcxfr ⊢ Ⅎ 𝑦 X 𝑥 ∈ 𝐴 𝐵