Theorem ixpeq1i

Description: Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypothesis	ixpeq1i.1	⊢ 𝐴 = 𝐵
Assertion	ixpeq1i	⊢ X 𝑥 ∈ 𝐴 𝐶 = X 𝑥 ∈ 𝐵 𝐶

Proof

Step	Hyp	Ref	Expression
1		ixpeq1i.1	⊢ 𝐴 = 𝐵
2	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
3	2	abbii	⊢ { 𝑥 ∣ 𝑥 ∈ 𝐴 } = { 𝑥 ∣ 𝑥 ∈ 𝐵 }
4	3	fneq2i	⊢ ( 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ↔ 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐵 } )
5	2	imbi1i	⊢ ( ( 𝑥 ∈ 𝐴 → ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 ) )
6	5	ralbii2	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 )
7	4 6	anbi12i	⊢ ( ( 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 ) ↔ ( 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐵 } ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 ) )
8	7	abbii	⊢ { 𝑓 ∣ ( 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 ) } = { 𝑓 ∣ ( 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐵 } ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 ) }
9		df-ixp	⊢ X 𝑥 ∈ 𝐴 𝐶 = { 𝑓 ∣ ( 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 ) }
10		df-ixp	⊢ X 𝑥 ∈ 𝐵 𝐶 = { 𝑓 ∣ ( 𝑓 Fn { 𝑥 ∣ 𝑥 ∈ 𝐵 } ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑓 ‘ 𝑥 ) ∈ 𝐶 ) }
11	8 9 10	3eqtr4i	⊢ X 𝑥 ∈ 𝐴 𝐶 = X 𝑥 ∈ 𝐵 𝐶