Theorem abfmpel

Description: Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016)

	Ref	Expression
Hypotheses	abfmpel.1	⊢ 𝐹 = ( 𝑥 ∈ 𝑉 ↦ { 𝑦 ∣ 𝜑 } )
	abfmpel.2	⊢ { 𝑦 ∣ 𝜑 } ∈ V
	abfmpel.3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	abfmpel	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		abfmpel.1	⊢ 𝐹 = ( 𝑥 ∈ 𝑉 ↦ { 𝑦 ∣ 𝜑 } )
2		abfmpel.2	⊢ { 𝑦 ∣ 𝜑 } ∈ V
3		abfmpel.3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
4	2	csbex	⊢ ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜑 } ∈ V
5	1	fvmpts	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜑 } ∈ V ) → ( 𝐹 ‘ 𝐴 ) = ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜑 } )
6	4 5	mpan2	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐹 ‘ 𝐴 ) = ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜑 } )
7		csbab	⊢ ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜑 } = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜑 }
8	6 7	eqtrdi	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐹 ‘ 𝐴 ) = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜑 } )
9	8	eleq2d	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜑 } ) )
10	9	adantr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜑 } ) )
11		simpl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵 ) → 𝐴 ∈ 𝑉 )
12	3	ancoms	⊢ ( ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐴 ) → ( 𝜑 ↔ 𝜓 ) )
13	12	adantll	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵 ) ∧ 𝑥 = 𝐴 ) → ( 𝜑 ↔ 𝜓 ) )
14	11 13	sbcied	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵 ) → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
15	14	ex	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑦 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 ) ) )
16	15	alrimiv	⊢ ( 𝐴 ∈ 𝑉 → ∀ 𝑦 ( 𝑦 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 ) ) )
17		elabgt	⊢ ( ( 𝐵 ∈ 𝑊 ∧ ∀ 𝑦 ( 𝑦 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 ) ) ) → ( 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜑 } ↔ 𝜓 ) )
18	16 17	sylan2	⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜑 } ↔ 𝜓 ) )
19	18	ancoms	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜑 } ↔ 𝜓 ) )
20	10 19	bitrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝜓 ) )