Theorem fnfocofob

Description: If the domain of a function G equals the range of a function F , then the composition ( G o. F ) is surjective iff G is surjective. (Contributed by GL and AV, 29-Sep-2024)

	Ref	Expression
Assertion	fnfocofob	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ ran 𝐹 = 𝐵 ) → ( ( 𝐺 ∘ 𝐹 ) : 𝐴 –onto→ 𝐶 ↔ 𝐺 : 𝐵 –onto→ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		cnvimarndm	⊢ ( ◡ 𝐹 “ ran 𝐹 ) = dom 𝐹
2		fndm	⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 )
3	2	3ad2ant1	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ ran 𝐹 = 𝐵 ) → dom 𝐹 = 𝐴 )
4	1 3	eqtr2id	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ ran 𝐹 = 𝐵 ) → 𝐴 = ( ◡ 𝐹 “ ran 𝐹 ) )
5		imaeq2	⊢ ( ran 𝐹 = 𝐵 → ( ◡ 𝐹 “ ran 𝐹 ) = ( ◡ 𝐹 “ 𝐵 ) )
6	5	3ad2ant3	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ ran 𝐹 = 𝐵 ) → ( ◡ 𝐹 “ ran 𝐹 ) = ( ◡ 𝐹 “ 𝐵 ) )
7	4 6	eqtrd	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ ran 𝐹 = 𝐵 ) → 𝐴 = ( ◡ 𝐹 “ 𝐵 ) )
8		foeq2	⊢ ( 𝐴 = ( ◡ 𝐹 “ 𝐵 ) → ( ( 𝐺 ∘ 𝐹 ) : 𝐴 –onto→ 𝐶 ↔ ( 𝐺 ∘ 𝐹 ) : ( ◡ 𝐹 “ 𝐵 ) –onto→ 𝐶 ) )
9	7 8	syl	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ ran 𝐹 = 𝐵 ) → ( ( 𝐺 ∘ 𝐹 ) : 𝐴 –onto→ 𝐶 ↔ ( 𝐺 ∘ 𝐹 ) : ( ◡ 𝐹 “ 𝐵 ) –onto→ 𝐶 ) )
10		fnfun	⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 )
11		id	⊢ ( 𝐺 : 𝐵 ⟶ 𝐶 → 𝐺 : 𝐵 ⟶ 𝐶 )
12		eqimss2	⊢ ( ran 𝐹 = 𝐵 → 𝐵 ⊆ ran 𝐹 )
13		funfocofob	⊢ ( ( Fun 𝐹 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ 𝐵 ⊆ ran 𝐹 ) → ( ( 𝐺 ∘ 𝐹 ) : ( ◡ 𝐹 “ 𝐵 ) –onto→ 𝐶 ↔ 𝐺 : 𝐵 –onto→ 𝐶 ) )
14	10 11 12 13	syl3an	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ ran 𝐹 = 𝐵 ) → ( ( 𝐺 ∘ 𝐹 ) : ( ◡ 𝐹 “ 𝐵 ) –onto→ 𝐶 ↔ 𝐺 : 𝐵 –onto→ 𝐶 ) )
15	9 14	bitrd	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ ran 𝐹 = 𝐵 ) → ( ( 𝐺 ∘ 𝐹 ) : 𝐴 –onto→ 𝐶 ↔ 𝐺 : 𝐵 –onto→ 𝐶 ) )