Metamath Proof Explorer


Theorem brimage

Description: Binary relation form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Hypotheses brimage.1 𝐴 ∈ V
brimage.2 𝐵 ∈ V
Assertion brimage ( 𝐴 Image 𝑅 𝐵𝐵 = ( 𝑅𝐴 ) )

Proof

Step Hyp Ref Expression
1 brimage.1 𝐴 ∈ V
2 brimage.2 𝐵 ∈ V
3 df-image Image 𝑅 = ( ( V × V ) ∖ ran ( ( V ⊗ E ) △ ( ( E ∘ 𝑅 ) ⊗ V ) ) )
4 brxp ( 𝐴 ( V × V ) 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
5 1 2 4 mpbir2an 𝐴 ( V × V ) 𝐵
6 vex 𝑥 ∈ V
7 vex 𝑦 ∈ V
8 6 7 brcnv ( 𝑥 𝑅 𝑦𝑦 𝑅 𝑥 )
9 8 rexbii ( ∃ 𝑦𝐴 𝑥 𝑅 𝑦 ↔ ∃ 𝑦𝐴 𝑦 𝑅 𝑥 )
10 6 1 coep ( 𝑥 ( E ∘ 𝑅 ) 𝐴 ↔ ∃ 𝑦𝐴 𝑥 𝑅 𝑦 )
11 6 elima ( 𝑥 ∈ ( 𝑅𝐴 ) ↔ ∃ 𝑦𝐴 𝑦 𝑅 𝑥 )
12 9 10 11 3bitr4ri ( 𝑥 ∈ ( 𝑅𝐴 ) ↔ 𝑥 ( E ∘ 𝑅 ) 𝐴 )
13 1 2 3 5 12 brtxpsd3 ( 𝐴 Image 𝑅 𝐵𝐵 = ( 𝑅𝐴 ) )