Theorem mptmpoopabbrd

Description: The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017) (Revised by AV, 15-Jan-2021)

	Ref	Expression
Hypotheses	mptmpoopabbrd.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
	mptmpoopabbrd.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐴 ‘ 𝐺 ) )
	mptmpoopabbrd.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ‘ 𝐺 ) )
	mptmpoopabbrd.v	⊢ ( 𝜑 → { ⟨ 𝑓 , ℎ ⟩ ∣ 𝜓 } ∈ 𝑉 )
	mptmpoopabbrd.r	⊢ ( ( 𝜑 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) → 𝜓 )
	mptmpoopabbrd.1	⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝑌 ) → ( 𝜏 ↔ 𝜃 ) )
	mptmpoopabbrd.2	⊢ ( 𝑔 = 𝐺 → ( 𝜒 ↔ 𝜏 ) )
	mptmpoopabbrd.m	⊢ 𝑀 = ( 𝑔 ∈ V ↦ ( 𝑎 ∈ ( 𝐴 ‘ 𝑔 ) , 𝑏 ∈ ( 𝐵 ‘ 𝑔 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜒 ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) } ) )
Assertion	mptmpoopabbrd	⊢ ( 𝜑 → ( 𝑋 ( 𝑀 ‘ 𝐺 ) 𝑌 ) = { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜃 ∧ 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		mptmpoopabbrd.v	⊢ ( 𝜑 → { ⟨ 𝑓 , ℎ ⟩ ∣ 𝜓 } ∈ 𝑉 )
5		mptmpoopabbrd.r	⊢ ( ( 𝜑 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) → 𝜓 )
6		mptmpoopabbrd.1	⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝑌 ) → ( 𝜏 ↔ 𝜃 ) )
7		mptmpoopabbrd.2	⊢ ( 𝑔 = 𝐺 → ( 𝜒 ↔ 𝜏 ) )
8		mptmpoopabbrd.m	⊢ 𝑀 = ( 𝑔 ∈ V ↦ ( 𝑎 ∈ ( 𝐴 ‘ 𝑔 ) , 𝑏 ∈ ( 𝐵 ‘ 𝑔 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜒 ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) } ) )
9		fveq2	⊢ ( 𝑔 = 𝐺 → ( 𝐴 ‘ 𝑔 ) = ( 𝐴 ‘ 𝐺 ) )
10		fveq2	⊢ ( 𝑔 = 𝐺 → ( 𝐵 ‘ 𝑔 ) = ( 𝐵 ‘ 𝐺 ) )
11		fveq2	⊢ ( 𝑔 = 𝐺 → ( 𝐷 ‘ 𝑔 ) = ( 𝐷 ‘ 𝐺 ) )
12	11	breqd	⊢ ( 𝑔 = 𝐺 → ( 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ↔ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) )
13	7 12	anbi12d	⊢ ( 𝑔 = 𝐺 → ( ( 𝜒 ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) ↔ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) ) )
14	13	opabbidv	⊢ ( 𝑔 = 𝐺 → { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜒 ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) } = { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } )
15	9 10 14	mpoeq123dv	⊢ ( 𝑔 = 𝐺 → ( 𝑎 ∈ ( 𝐴 ‘ 𝑔 ) , 𝑏 ∈ ( 𝐵 ‘ 𝑔 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜒 ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) } ) = ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) )
16		elex	⊢ ( 𝐺 ∈ 𝑊 → 𝐺 ∈ V )
17	16	adantr	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐺 ∈ 𝑊 ) → 𝐺 ∈ V )
18		fvex	⊢ ( 𝐴 ‘ 𝐺 ) ∈ V
19		fvex	⊢ ( 𝐵 ‘ 𝐺 ) ∈ V
20	18 19	pm3.2i	⊢ ( ( 𝐴 ‘ 𝐺 ) ∈ V ∧ ( 𝐵 ‘ 𝐺 ) ∈ V )
21		mpoexga	⊢ ( ( ( 𝐴 ‘ 𝐺 ) ∈ V ∧ ( 𝐵 ‘ 𝐺 ) ∈ V ) → ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) ∈ V )
22	20 21	mp1i	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐺 ∈ 𝑊 ) → ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) ∈ V )
23	8 15 17 22	fvmptd3	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐺 ∈ 𝑊 ) → ( 𝑀 ‘ 𝐺 ) = ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) )
24	1 1 23	syl2anc	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐺 ) = ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) )
25	24	oveqd	⊢ ( 𝜑 → ( 𝑋 ( 𝑀 ‘ 𝐺 ) 𝑌 ) = ( 𝑋 ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) 𝑌 ) )
26		ancom	⊢ ( ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) ↔ ( 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ∧ 𝜃 ) )
27	26	opabbii	⊢ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } = { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ∧ 𝜃 ) }
28	5 4	opabresex2d	⊢ ( 𝜑 → { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ∧ 𝜃 ) } ∈ V )
29	27 28	eqeltrid	⊢ ( 𝜑 → { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ∈ V )
30	6	anbi1d	⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝑌 ) → ( ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) ↔ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) ) )
31	30	opabbidv	⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝑌 ) → { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } = { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } )
32		eqid	⊢ ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) = ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } )
33	31 32	ovmpoga	⊢ ( ( 𝑋 ∈ ( 𝐴 ‘ 𝐺 ) ∧ 𝑌 ∈ ( 𝐵 ‘ 𝐺 ) ∧ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ∈ V ) → ( 𝑋 ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) 𝑌 ) = { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } )
34	2 3 29 33	syl3anc	⊢ ( 𝜑 → ( 𝑋 ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) 𝑌 ) = { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } )
35	25 34	eqtrd	⊢ ( 𝜑 → ( 𝑋 ( 𝑀 ‘ 𝐺 ) 𝑌 ) = { ⟨ 𝑓 , ℎ ⟩ ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } )