Metamath Proof Explorer


Theorem offveq

Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014)

Ref Expression
Hypotheses offveq.1 ( 𝜑𝐴𝑉 )
offveq.2 ( 𝜑𝐹 Fn 𝐴 )
offveq.3 ( 𝜑𝐺 Fn 𝐴 )
offveq.4 ( 𝜑𝐻 Fn 𝐴 )
offveq.5 ( ( 𝜑𝑥𝐴 ) → ( 𝐹𝑥 ) = 𝐵 )
offveq.6 ( ( 𝜑𝑥𝐴 ) → ( 𝐺𝑥 ) = 𝐶 )
offveq.7 ( ( 𝜑𝑥𝐴 ) → ( 𝐵 𝑅 𝐶 ) = ( 𝐻𝑥 ) )
Assertion offveq ( 𝜑 → ( 𝐹f 𝑅 𝐺 ) = 𝐻 )

Proof

Step Hyp Ref Expression
1 offveq.1 ( 𝜑𝐴𝑉 )
2 offveq.2 ( 𝜑𝐹 Fn 𝐴 )
3 offveq.3 ( 𝜑𝐺 Fn 𝐴 )
4 offveq.4 ( 𝜑𝐻 Fn 𝐴 )
5 offveq.5 ( ( 𝜑𝑥𝐴 ) → ( 𝐹𝑥 ) = 𝐵 )
6 offveq.6 ( ( 𝜑𝑥𝐴 ) → ( 𝐺𝑥 ) = 𝐶 )
7 offveq.7 ( ( 𝜑𝑥𝐴 ) → ( 𝐵 𝑅 𝐶 ) = ( 𝐻𝑥 ) )
8 inidm ( 𝐴𝐴 ) = 𝐴
9 2 3 1 1 8 offn ( 𝜑 → ( 𝐹f 𝑅 𝐺 ) Fn 𝐴 )
10 2 3 1 1 8 5 6 ofval ( ( 𝜑𝑥𝐴 ) → ( ( 𝐹f 𝑅 𝐺 ) ‘ 𝑥 ) = ( 𝐵 𝑅 𝐶 ) )
11 10 7 eqtrd ( ( 𝜑𝑥𝐴 ) → ( ( 𝐹f 𝑅 𝐺 ) ‘ 𝑥 ) = ( 𝐻𝑥 ) )
12 9 4 11 eqfnfvd ( 𝜑 → ( 𝐹f 𝑅 𝐺 ) = 𝐻 )