Theorem csbmpt12

Description: Move substitution into a maps-to notation. (Contributed by AV, 26-Sep-2019)

	Ref	Expression
Assertion	csbmpt12	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝑦 ∈ 𝑌 ↦ 𝑍 ) = ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ↦ ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		csbopab	⊢ ⦋ 𝐴 / 𝑥 ⦌ { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍 ) } = { ⟨ 𝑦 , 𝑧 ⟩ ∣ [ 𝐴 / 𝑥 ] ( 𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍 ) }
2		sbcan	⊢ ( [ 𝐴 / 𝑥 ] ( 𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍 ) ↔ ( [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝑌 ∧ [ 𝐴 / 𝑥 ] 𝑧 = 𝑍 ) )
3		sbcel12	⊢ ( [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝑌 ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 )
4		csbconstg	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ 𝑦 = 𝑦 )
5	4	eleq1d	⊢ ( 𝐴 ∈ 𝑉 → ( ⦋ 𝐴 / 𝑥 ⦌ 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ↔ 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ) )
6	3 5	bitrid	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝑌 ↔ 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ) )
7		sbceq2g	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝑧 = 𝑍 ↔ 𝑧 = ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) )
8	6 7	anbi12d	⊢ ( 𝐴 ∈ 𝑉 → ( ( [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝑌 ∧ [ 𝐴 / 𝑥 ] 𝑧 = 𝑍 ) ↔ ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ∧ 𝑧 = ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) ) )
9	2 8	bitrid	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍 ) ↔ ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ∧ 𝑧 = ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) ) )
10	9	opabbidv	⊢ ( 𝐴 ∈ 𝑉 → { ⟨ 𝑦 , 𝑧 ⟩ ∣ [ 𝐴 / 𝑥 ] ( 𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍 ) } = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ∧ 𝑧 = ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) } )
11	1 10	eqtrid	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍 ) } = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ∧ 𝑧 = ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) } )
12		df-mpt	⊢ ( 𝑦 ∈ 𝑌 ↦ 𝑍 ) = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍 ) }
13	12	csbeq2i	⊢ ⦋ 𝐴 / 𝑥 ⦌ ( 𝑦 ∈ 𝑌 ↦ 𝑍 ) = ⦋ 𝐴 / 𝑥 ⦌ { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍 ) }
14		df-mpt	⊢ ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ↦ ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ∧ 𝑧 = ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) }
15	11 13 14	3eqtr4g	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝑦 ∈ 𝑌 ↦ 𝑍 ) = ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ↦ ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) )