Metamath Proof Explorer


Theorem coepr

Description: Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013)

Ref Expression
Hypotheses coep.1 𝐴 ∈ V
coep.2 𝐵 ∈ V
Assertion coepr ( 𝐴 ( 𝑅 E ) 𝐵 ↔ ∃ 𝑥𝐴 𝑥 𝑅 𝐵 )

Proof

Step Hyp Ref Expression
1 coep.1 𝐴 ∈ V
2 coep.2 𝐵 ∈ V
3 vex 𝑥 ∈ V
4 1 3 brcnv ( 𝐴 E 𝑥𝑥 E 𝐴 )
5 1 epeli ( 𝑥 E 𝐴𝑥𝐴 )
6 4 5 bitri ( 𝐴 E 𝑥𝑥𝐴 )
7 6 anbi1i ( ( 𝐴 E 𝑥𝑥 𝑅 𝐵 ) ↔ ( 𝑥𝐴𝑥 𝑅 𝐵 ) )
8 7 exbii ( ∃ 𝑥 ( 𝐴 E 𝑥𝑥 𝑅 𝐵 ) ↔ ∃ 𝑥 ( 𝑥𝐴𝑥 𝑅 𝐵 ) )
9 1 2 brco ( 𝐴 ( 𝑅 E ) 𝐵 ↔ ∃ 𝑥 ( 𝐴 E 𝑥𝑥 𝑅 𝐵 ) )
10 df-rex ( ∃ 𝑥𝐴 𝑥 𝑅 𝐵 ↔ ∃ 𝑥 ( 𝑥𝐴𝑥 𝑅 𝐵 ) )
11 8 9 10 3bitr4i ( 𝐴 ( 𝑅 E ) 𝐵 ↔ ∃ 𝑥𝐴 𝑥 𝑅 𝐵 )