Metamath Proof Explorer

Theorem ofrco

Description: Function relation between function compositions. (Contributed by Thierry Arnoux, 15-Jan-2026)

Ref Expression
Hypotheses ofrco.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
ofrco.2 ⊢ ( 𝜑 → 𝐺 Fn 𝐴 )
ofrco.3 ⊢ ( 𝜑 → 𝐻 : 𝐶 ⟶ 𝐴 )
ofrco.4 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
ofrco.5 ⊢ ( 𝜑 → 𝐶 ∈ 𝑊 )
ofrco.6 ⊢ ( 𝜑 → 𝐹 ∘r 𝑅 𝐺 )
Assertion ofrco ( 𝜑 → ( 𝐹 ∘ 𝐻 ) ∘r 𝑅 ( 𝐺 ∘ 𝐻 ) )

Proof

Step Hyp Ref Expression
1 ofrco.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
2 ofrco.2 ⊢ ( 𝜑 → 𝐺 Fn 𝐴 )
3 ofrco.3 ⊢ ( 𝜑 → 𝐻 : 𝐶 ⟶ 𝐴 )
4 ofrco.4 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
5 ofrco.5 ⊢ ( 𝜑 → 𝐶 ∈ 𝑊 )
6 ofrco.6 ⊢ ( 𝜑 → 𝐹 ∘r 𝑅 𝐺 )
7 fveq2 ⊢ ( 𝑦 = ( 𝐻 ‘ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝐻 ‘ 𝑥 ) ) )
8 fveq2 ⊢ ( 𝑦 = ( 𝐻 ‘ 𝑥 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐻 ‘ 𝑥 ) ) )
9 7 8 breq12d ⊢ ( 𝑦 = ( 𝐻 ‘ 𝑥 ) → ( ( 𝐹 ‘ 𝑦 ) 𝑅 ( 𝐺 ‘ 𝑦 ) ↔ ( 𝐹 ‘ ( 𝐻 ‘ 𝑥 ) ) 𝑅 ( 𝐺 ‘ ( 𝐻 ‘ 𝑥 ) ) ) )
10 inidm ⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴
11 eqidd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
12 eqidd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
13 1 2 4 4 10 11 12 ofrfval ⊢ ( 𝜑 → ( 𝐹 ∘r 𝑅 𝐺 ↔ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) 𝑅 ( 𝐺 ‘ 𝑦 ) ) )
14 6 13 mpbid ⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) 𝑅 ( 𝐺 ‘ 𝑦 ) )
15 14 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) 𝑅 ( 𝐺 ‘ 𝑦 ) )
16 3 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ( 𝐻 ‘ 𝑥 ) ∈ 𝐴 )
17 9 15 16 rspcdva ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ( 𝐹 ‘ ( 𝐻 ‘ 𝑥 ) ) 𝑅 ( 𝐺 ‘ ( 𝐻 ‘ 𝑥 ) ) )
18 17 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐶 ( 𝐹 ‘ ( 𝐻 ‘ 𝑥 ) ) 𝑅 ( 𝐺 ‘ ( 𝐻 ‘ 𝑥 ) ) )
19 fnfco ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐻 : 𝐶 ⟶ 𝐴 ) → ( 𝐹 ∘ 𝐻 ) Fn 𝐶 )
20 1 3 19 syl2anc ⊢ ( 𝜑 → ( 𝐹 ∘ 𝐻 ) Fn 𝐶 )
21 fnfco ⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝐻 : 𝐶 ⟶ 𝐴 ) → ( 𝐺 ∘ 𝐻 ) Fn 𝐶 )
22 2 3 21 syl2anc ⊢ ( 𝜑 → ( 𝐺 ∘ 𝐻 ) Fn 𝐶 )
23 inidm ⊢ ( 𝐶 ∩ 𝐶 ) = 𝐶
24 3 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝐻 : 𝐶 ⟶ 𝐴 )
25 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝑥 ∈ 𝐶 )
26 24 25 fvco3d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝐻 ‘ 𝑥 ) ) )
27 24 25 fvco3d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ( ( 𝐺 ∘ 𝐻 ) ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝐻 ‘ 𝑥 ) ) )
28 20 22 5 5 23 26 27 ofrfval ⊢ ( 𝜑 → ( ( 𝐹 ∘ 𝐻 ) ∘r 𝑅 ( 𝐺 ∘ 𝐻 ) ↔ ∀ 𝑥 ∈ 𝐶 ( 𝐹 ‘ ( 𝐻 ‘ 𝑥 ) ) 𝑅 ( 𝐺 ‘ ( 𝐻 ‘ 𝑥 ) ) ) )
29 18 28 mpbird ⊢ ( 𝜑 → ( 𝐹 ∘ 𝐻 ) ∘r 𝑅 ( 𝐺 ∘ 𝐻 ) )