Metamath Proof Explorer


Theorem ofcfval4

Description: The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017)

Ref Expression
Hypotheses ofcfval4.1 ( 𝜑𝐹 : 𝐴𝐵 )
ofcfval4.2 ( 𝜑𝐴𝑉 )
ofcfval4.3 ( 𝜑𝐶𝑊 )
Assertion ofcfval4 ( 𝜑 → ( 𝐹f/c 𝑅 𝐶 ) = ( ( 𝑥𝐵 ↦ ( 𝑥 𝑅 𝐶 ) ) ∘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 ofcfval4.1 ( 𝜑𝐹 : 𝐴𝐵 )
2 ofcfval4.2 ( 𝜑𝐴𝑉 )
3 ofcfval4.3 ( 𝜑𝐶𝑊 )
4 1 fdmd ( 𝜑 → dom 𝐹 = 𝐴 )
5 4 mpteq1d ( 𝜑 → ( 𝑦 ∈ dom 𝐹 ↦ ( ( 𝐹𝑦 ) 𝑅 𝐶 ) ) = ( 𝑦𝐴 ↦ ( ( 𝐹𝑦 ) 𝑅 𝐶 ) ) )
6 1 2 fexd ( 𝜑𝐹 ∈ V )
7 ofcfval3 ( ( 𝐹 ∈ V ∧ 𝐶𝑊 ) → ( 𝐹f/c 𝑅 𝐶 ) = ( 𝑦 ∈ dom 𝐹 ↦ ( ( 𝐹𝑦 ) 𝑅 𝐶 ) ) )
8 6 3 7 syl2anc ( 𝜑 → ( 𝐹f/c 𝑅 𝐶 ) = ( 𝑦 ∈ dom 𝐹 ↦ ( ( 𝐹𝑦 ) 𝑅 𝐶 ) ) )
9 1 ffvelrnda ( ( 𝜑𝑦𝐴 ) → ( 𝐹𝑦 ) ∈ 𝐵 )
10 1 feqmptd ( 𝜑𝐹 = ( 𝑦𝐴 ↦ ( 𝐹𝑦 ) ) )
11 eqidd ( 𝜑 → ( 𝑥𝐵 ↦ ( 𝑥 𝑅 𝐶 ) ) = ( 𝑥𝐵 ↦ ( 𝑥 𝑅 𝐶 ) ) )
12 oveq1 ( 𝑥 = ( 𝐹𝑦 ) → ( 𝑥 𝑅 𝐶 ) = ( ( 𝐹𝑦 ) 𝑅 𝐶 ) )
13 9 10 11 12 fmptco ( 𝜑 → ( ( 𝑥𝐵 ↦ ( 𝑥 𝑅 𝐶 ) ) ∘ 𝐹 ) = ( 𝑦𝐴 ↦ ( ( 𝐹𝑦 ) 𝑅 𝐶 ) ) )
14 5 8 13 3eqtr4d ( 𝜑 → ( 𝐹f/c 𝑅 𝐶 ) = ( ( 𝑥𝐵 ↦ ( 𝑥 𝑅 𝐶 ) ) ∘ 𝐹 ) )