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Metamath Proof Explorer

Theorem f1ocoima

Description: The composition of two bijections as bijection onto the image of the range of the first bijection. (Contributed by AV, 15-Aug-2025)

Ref Expression
Assertion f1ocoima ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 f1of1 ⊢ ( 𝐺 : 𝐶 –1-1-onto→ 𝐷 → 𝐺 : 𝐶 –1-1→ 𝐷 )
2 1 anim1i ⊢ ( ( 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 : 𝐶 –1-1→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) )
3 2 3adant1 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 : 𝐶 –1-1→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) )
4 f1ores ⊢ ( ( 𝐺 : 𝐶 –1-1→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 ↾ 𝐵 ) : 𝐵 –1-1-onto→ ( 𝐺 “ 𝐵 ) )
5 3 4 syl ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 ↾ 𝐵 ) : 𝐵 –1-1-onto→ ( 𝐺 “ 𝐵 ) )
6 simp1 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → 𝐹 : 𝐴 –1-1-onto→ 𝐵 )
7 f1oco ⊢ ( ( ( 𝐺 ↾ 𝐵 ) : 𝐵 –1-1-onto→ ( 𝐺 “ 𝐵 ) ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) )
8 5 6 7 syl2anc ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) )
9 f1ofo ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 : 𝐴 –onto→ 𝐵 )
10 forn ⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ran 𝐹 = 𝐵 )
11 9 10 syl ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ran 𝐹 = 𝐵 )
12 11 eqimssd ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ran 𝐹 ⊆ 𝐵 )
13 12 3ad2ant1 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ran 𝐹 ⊆ 𝐵 )
14 cores ⊢ ( ran 𝐹 ⊆ 𝐵 → ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) = ( 𝐺 ∘ 𝐹 ) )
15 14 eqcomd ⊢ ( ran 𝐹 ⊆ 𝐵 → ( 𝐺 ∘ 𝐹 ) = ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) )
16 13 15 syl ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 ∘ 𝐹 ) = ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) )
17 16 f1oeq1d ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( ( 𝐺 ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) ↔ ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) ) )
18 8 17 mpbird ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) )