Theorem f1ocnvd

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Hypotheses	f1od.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )
	f1od.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑊 )
	f1od.3	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ 𝑋 )
	f1od.4	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷 ) ) )
Assertion	f1ocnvd	⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ ◡ 𝐹 = ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		f1od.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )
2		f1od.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑊 )
3		f1od.3	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ 𝑋 )
4		f1od.4	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷 ) ) )
5	2	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝑊 )
6	1	fnmpt	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝑊 → 𝐹 Fn 𝐴 )
7	5 6	syl	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
8	3	ralrimiva	⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 𝐷 ∈ 𝑋 )
9		eqid	⊢ ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) = ( 𝑦 ∈ 𝐵 ↦ 𝐷 )
10	9	fnmpt	⊢ ( ∀ 𝑦 ∈ 𝐵 𝐷 ∈ 𝑋 → ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) Fn 𝐵 )
11	8 10	syl	⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) Fn 𝐵 )
12	4	opabbidv	⊢ ( 𝜑 → { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) } = { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷 ) } )
13		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) }
14	1 13	eqtri	⊢ 𝐹 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) }
15	14	cnveqi	⊢ ◡ 𝐹 = ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) }
16		cnvopab	⊢ ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) } = { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) }
17	15 16	eqtri	⊢ ◡ 𝐹 = { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) }
18		df-mpt	⊢ ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) = { ⟨ 𝑦 , 𝑥 ⟩ ∣ ( 𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷 ) }
19	12 17 18	3eqtr4g	⊢ ( 𝜑 → ◡ 𝐹 = ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) )
20	19	fneq1d	⊢ ( 𝜑 → ( ◡ 𝐹 Fn 𝐵 ↔ ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) Fn 𝐵 ) )
21	11 20	mpbird	⊢ ( 𝜑 → ◡ 𝐹 Fn 𝐵 )
22		dff1o4	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ◡ 𝐹 Fn 𝐵 ) )
23	7 21 22	sylanbrc	⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1-onto→ 𝐵 )
24	23 19	jca	⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ ◡ 𝐹 = ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) ) )