Theorem abfmpeld

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

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

Proof

Step	Hyp	Ref	Expression
1		abfmpeld.1	⊢ 𝐹 = ( 𝑥 ∈ 𝑉 ↦ { 𝑦 ∣ 𝜓 } )
2		abfmpeld.2	⊢ ( 𝜑 → { 𝑦 ∣ 𝜓 } ∈ V )
3		abfmpeld.3	⊢ ( 𝜑 → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜓 ↔ 𝜒 ) ) )
4	2	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 { 𝑦 ∣ 𝜓 } ∈ V )
5		csbexg	⊢ ( ∀ 𝑥 { 𝑦 ∣ 𝜓 } ∈ V → ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } ∈ V )
6	4 5	syl	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } ∈ V )
7	1	fvmpts	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } ∈ V ) → ( 𝐹 ‘ 𝐴 ) = ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } )
8	6 7	sylan2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( 𝐹 ‘ 𝐴 ) = ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } )
9		csbab	⊢ ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 }
10	8 9	eqtrdi	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( 𝐹 ‘ 𝐴 ) = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } )
11	10	eleq2d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } ) )
12	11	adantl	⊢ ( ( 𝐵 ∈ 𝑊 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } ) )
13		simpll	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ∧ 𝑦 = 𝐵 ) → 𝐴 ∈ 𝑉 )
14	3	ancomsd	⊢ ( 𝜑 → ( ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) )
15	14	adantl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) )
16	15	impl	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ∧ 𝑦 = 𝐵 ) ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
17	13 16	sbcied	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ∧ 𝑦 = 𝐵 ) → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) )
18	17	ex	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑦 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) )
19	18	alrimiv	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ∀ 𝑦 ( 𝑦 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) )
20		elabgt	⊢ ( ( 𝐵 ∈ 𝑊 ∧ ∀ 𝑦 ( 𝑦 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) ) → ( 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } ↔ 𝜒 ) )
21	19 20	sylan2	⊢ ( ( 𝐵 ∈ 𝑊 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ) → ( 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } ↔ 𝜒 ) )
22	12 21	bitrd	⊢ ( ( 𝐵 ∈ 𝑊 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝜒 ) )
23	22	an13s	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝜒 ) )
24	23	ex	⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝜒 ) ) )