Theorem finxpeq2

Description: Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020)

	Ref	Expression
Assertion	finxpeq2	⊢ ( 𝑀 = 𝑁 → ( 𝑈 ↑↑ 𝑀 ) = ( 𝑈 ↑↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝑀 = 𝑁 → ( 𝑀 ∈ ω ↔ 𝑁 ∈ ω ) )
2		opeq1	⊢ ( 𝑀 = 𝑁 → ⟨ 𝑀 , 𝑦 ⟩ = ⟨ 𝑁 , 𝑦 ⟩ )
3		rdgeq2	⊢ ( ⟨ 𝑀 , 𝑦 ⟩ = ⟨ 𝑁 , 𝑦 ⟩ → rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑀 , 𝑦 ⟩ ) = rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑁 , 𝑦 ⟩ ) )
4	2 3	syl	⊢ ( 𝑀 = 𝑁 → rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑀 , 𝑦 ⟩ ) = rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑁 , 𝑦 ⟩ ) )
5		id	⊢ ( 𝑀 = 𝑁 → 𝑀 = 𝑁 )
6	4 5	fveq12d	⊢ ( 𝑀 = 𝑁 → ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑀 , 𝑦 ⟩ ) ‘ 𝑀 ) = ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑁 , 𝑦 ⟩ ) ‘ 𝑁 ) )
7	6	eqeq2d	⊢ ( 𝑀 = 𝑁 → ( ∅ = ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑀 , 𝑦 ⟩ ) ‘ 𝑀 ) ↔ ∅ = ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑁 , 𝑦 ⟩ ) ‘ 𝑁 ) ) )
8	1 7	anbi12d	⊢ ( 𝑀 = 𝑁 → ( ( 𝑀 ∈ ω ∧ ∅ = ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑀 , 𝑦 ⟩ ) ‘ 𝑀 ) ) ↔ ( 𝑁 ∈ ω ∧ ∅ = ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑁 , 𝑦 ⟩ ) ‘ 𝑁 ) ) ) )
9	8	abbidv	⊢ ( 𝑀 = 𝑁 → { 𝑦 ∣ ( 𝑀 ∈ ω ∧ ∅ = ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑀 , 𝑦 ⟩ ) ‘ 𝑀 ) ) } = { 𝑦 ∣ ( 𝑁 ∈ ω ∧ ∅ = ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑁 , 𝑦 ⟩ ) ‘ 𝑁 ) ) } )
10		df-finxp	⊢ ( 𝑈 ↑↑ 𝑀 ) = { 𝑦 ∣ ( 𝑀 ∈ ω ∧ ∅ = ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑀 , 𝑦 ⟩ ) ‘ 𝑀 ) ) }
11		df-finxp	⊢ ( 𝑈 ↑↑ 𝑁 ) = { 𝑦 ∣ ( 𝑁 ∈ ω ∧ ∅ = ( rec ( ( 𝑛 ∈ ω , 𝑥 ∈ V ↦ if ( ( 𝑛 = 1o ∧ 𝑥 ∈ 𝑈 ) , ∅ , if ( 𝑥 ∈ ( V × 𝑈 ) , ⟨ ∪ 𝑛 , ( 1st ‘ 𝑥 ) ⟩ , ⟨ 𝑛 , 𝑥 ⟩ ) ) ) , ⟨ 𝑁 , 𝑦 ⟩ ) ‘ 𝑁 ) ) }
12	9 10 11	3eqtr4g	⊢ ( 𝑀 = 𝑁 → ( 𝑈 ↑↑ 𝑀 ) = ( 𝑈 ↑↑ 𝑁 ) )