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