Metamath Proof Explorer


Definition df-ofr

Description: Define the function relation map. The definition is designed so that if R is a binary relation, then oR R is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014)

Ref Expression
Assertion df-ofr r 𝑅 = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ∀ 𝑥 ∈ ( dom 𝑓 ∩ dom 𝑔 ) ( 𝑓𝑥 ) 𝑅 ( 𝑔𝑥 ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cR 𝑅
1 0 cofr r 𝑅
2 vf 𝑓
3 vg 𝑔
4 vx 𝑥
5 2 cv 𝑓
6 5 cdm dom 𝑓
7 3 cv 𝑔
8 7 cdm dom 𝑔
9 6 8 cin ( dom 𝑓 ∩ dom 𝑔 )
10 4 cv 𝑥
11 10 5 cfv ( 𝑓𝑥 )
12 10 7 cfv ( 𝑔𝑥 )
13 11 12 0 wbr ( 𝑓𝑥 ) 𝑅 ( 𝑔𝑥 )
14 13 4 9 wral 𝑥 ∈ ( dom 𝑓 ∩ dom 𝑔 ) ( 𝑓𝑥 ) 𝑅 ( 𝑔𝑥 )
15 14 2 3 copab { ⟨ 𝑓 , 𝑔 ⟩ ∣ ∀ 𝑥 ∈ ( dom 𝑓 ∩ dom 𝑔 ) ( 𝑓𝑥 ) 𝑅 ( 𝑔𝑥 ) }
16 1 15 wceq r 𝑅 = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ∀ 𝑥 ∈ ( dom 𝑓 ∩ dom 𝑔 ) ( 𝑓𝑥 ) 𝑅 ( 𝑔𝑥 ) }