Metamath Proof Explorer


Theorem ofc12

Description: Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014)

Ref Expression
Hypotheses ofc12.1 ( 𝜑𝐴𝑉 )
ofc12.2 ( 𝜑𝐵𝑊 )
ofc12.3 ( 𝜑𝐶𝑋 )
Assertion ofc12 ( 𝜑 → ( ( 𝐴 × { 𝐵 } ) ∘f 𝑅 ( 𝐴 × { 𝐶 } ) ) = ( 𝐴 × { ( 𝐵 𝑅 𝐶 ) } ) )

Proof

Step Hyp Ref Expression
1 ofc12.1 ( 𝜑𝐴𝑉 )
2 ofc12.2 ( 𝜑𝐵𝑊 )
3 ofc12.3 ( 𝜑𝐶𝑋 )
4 2 adantr ( ( 𝜑𝑥𝐴 ) → 𝐵𝑊 )
5 3 adantr ( ( 𝜑𝑥𝐴 ) → 𝐶𝑋 )
6 fconstmpt ( 𝐴 × { 𝐵 } ) = ( 𝑥𝐴𝐵 )
7 6 a1i ( 𝜑 → ( 𝐴 × { 𝐵 } ) = ( 𝑥𝐴𝐵 ) )
8 fconstmpt ( 𝐴 × { 𝐶 } ) = ( 𝑥𝐴𝐶 )
9 8 a1i ( 𝜑 → ( 𝐴 × { 𝐶 } ) = ( 𝑥𝐴𝐶 ) )
10 1 4 5 7 9 offval2 ( 𝜑 → ( ( 𝐴 × { 𝐵 } ) ∘f 𝑅 ( 𝐴 × { 𝐶 } ) ) = ( 𝑥𝐴 ↦ ( 𝐵 𝑅 𝐶 ) ) )
11 fconstmpt ( 𝐴 × { ( 𝐵 𝑅 𝐶 ) } ) = ( 𝑥𝐴 ↦ ( 𝐵 𝑅 𝐶 ) )
12 10 11 eqtr4di ( 𝜑 → ( ( 𝐴 × { 𝐵 } ) ∘f 𝑅 ( 𝐴 × { 𝐶 } ) ) = ( 𝐴 × { ( 𝐵 𝑅 𝐶 ) } ) )