Theorem mptmpoopabovd

Description: The operation value of a function value of a collection of ordered pairs of related elements. (Contributed by Alexander van der Vekens, 8-Nov-2017) (Revised by AV, 15-Jan-2021) Add disjoint variable condition on D , f , h to remove hypotheses. (Revised by SN, 13-Dec-2024)

	Ref	Expression
Hypotheses	mptmpoopabbrd.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
	mptmpoopabbrd.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐴 ‘ 𝐺 ) )
	mptmpoopabbrd.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ‘ 𝐺 ) )
	mptmpoopabovd.m	⊢ 𝑀 = ( 𝑔 ∈ V ↦ ( 𝑎 ∈ ( 𝐴 ‘ 𝑔 ) , 𝑏 ∈ ( 𝐵 ‘ 𝑔 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝑓 ( 𝑎 ( 𝐶 ‘ 𝑔 ) 𝑏 ) ℎ ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) } ) )
Assertion	mptmpoopabovd	⊢ ( 𝜑 → ( 𝑋 ( 𝑀 ‘ 𝐺 ) 𝑌 ) = { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝑓 ( 𝑋 ( 𝐶 ‘ 𝐺 ) 𝑌 ) ℎ ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } )

Proof

Step	Hyp	Ref	Expression
1		mptmpoopabbrd.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
2		mptmpoopabbrd.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐴 ‘ 𝐺 ) )
3		mptmpoopabbrd.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ‘ 𝐺 ) )
4		mptmpoopabovd.m	⊢ 𝑀 = ( 𝑔 ∈ V ↦ ( 𝑎 ∈ ( 𝐴 ‘ 𝑔 ) , 𝑏 ∈ ( 𝐵 ‘ 𝑔 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝑓 ( 𝑎 ( 𝐶 ‘ 𝑔 ) 𝑏 ) ℎ ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) } ) )
5		oveq12	⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝑌 ) → ( 𝑎 ( 𝐶 ‘ 𝐺 ) 𝑏 ) = ( 𝑋 ( 𝐶 ‘ 𝐺 ) 𝑌 ) )
6	5	breqd	⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝑌 ) → ( 𝑓 ( 𝑎 ( 𝐶 ‘ 𝐺 ) 𝑏 ) ℎ ↔ 𝑓 ( 𝑋 ( 𝐶 ‘ 𝐺 ) 𝑌 ) ℎ ) )
7		fveq2	⊢ ( 𝑔 = 𝐺 → ( 𝐶 ‘ 𝑔 ) = ( 𝐶 ‘ 𝐺 ) )
8	7	oveqd	⊢ ( 𝑔 = 𝐺 → ( 𝑎 ( 𝐶 ‘ 𝑔 ) 𝑏 ) = ( 𝑎 ( 𝐶 ‘ 𝐺 ) 𝑏 ) )
9	8	breqd	⊢ ( 𝑔 = 𝐺 → ( 𝑓 ( 𝑎 ( 𝐶 ‘ 𝑔 ) 𝑏 ) ℎ ↔ 𝑓 ( 𝑎 ( 𝐶 ‘ 𝐺 ) 𝑏 ) ℎ ) )
10	1 2 3 6 9 4	mptmpoopabbrd	⊢ ( 𝜑 → ( 𝑋 ( 𝑀 ‘ 𝐺 ) 𝑌 ) = { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝑓 ( 𝑋 ( 𝐶 ‘ 𝐺 ) 𝑌 ) ℎ ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } )