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Metamath Proof Explorer

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 ) ) )