Theorem focofob

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 and F as function to the domain of G are both surjective. Symmetric version of fnfocofob including the fact that F is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024) (Proof shortened by AV, 29-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ffn	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 Fn 𝐴 )
2		fnfocofob	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 : 𝐶 ⟶ 𝐷 ∧ ran 𝐹 = 𝐶 ) → ( ( 𝐺 ∘ 𝐹 ) : 𝐴 –onto→ 𝐷 ↔ 𝐺 : 𝐶 –onto→ 𝐷 ) )
3	1 2	syl3an1	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 : 𝐶 ⟶ 𝐷 ∧ ran 𝐹 = 𝐶 ) → ( ( 𝐺 ∘ 𝐹 ) : 𝐴 –onto→ 𝐷 ↔ 𝐺 : 𝐶 –onto→ 𝐷 ) )
4		dffn4	⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 –onto→ ran 𝐹 )
5	1 4	sylib	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 : 𝐴 –onto→ ran 𝐹 )
6	5	3ad2ant1	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 : 𝐶 ⟶ 𝐷 ∧ ran 𝐹 = 𝐶 ) → 𝐹 : 𝐴 –onto→ ran 𝐹 )
7		foeq3	⊢ ( ran 𝐹 = 𝐶 → ( 𝐹 : 𝐴 –onto→ ran 𝐹 ↔ 𝐹 : 𝐴 –onto→ 𝐶 ) )
8	7	3ad2ant3	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 : 𝐶 ⟶ 𝐷 ∧ ran 𝐹 = 𝐶 ) → ( 𝐹 : 𝐴 –onto→ ran 𝐹 ↔ 𝐹 : 𝐴 –onto→ 𝐶 ) )
9	6 8	mpbid	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 : 𝐶 ⟶ 𝐷 ∧ ran 𝐹 = 𝐶 ) → 𝐹 : 𝐴 –onto→ 𝐶 )
10	9	biantrurd	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 : 𝐶 ⟶ 𝐷 ∧ ran 𝐹 = 𝐶 ) → ( 𝐺 : 𝐶 –onto→ 𝐷 ↔ ( 𝐹 : 𝐴 –onto→ 𝐶 ∧ 𝐺 : 𝐶 –onto→ 𝐷 ) ) )
11	3 10	bitrd	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 : 𝐶 ⟶ 𝐷 ∧ ran 𝐹 = 𝐶 ) → ( ( 𝐺 ∘ 𝐹 ) : 𝐴 –onto→ 𝐷 ↔ ( 𝐹 : 𝐴 –onto→ 𝐶 ∧ 𝐺 : 𝐶 –onto→ 𝐷 ) ) )