Metamath Proof Explorer


Theorem brfvidRP

Description: If two elements are connected by a value of the identity relation, then they are connected via the argument. This is an example which uses brmptiunrelexpd . (Contributed by RP, 21-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypothesis brfvidRP.r ( 𝜑𝑅 ∈ V )
Assertion brfvidRP ( 𝜑 → ( 𝐴 ( I ‘ 𝑅 ) 𝐵𝐴 𝑅 𝐵 ) )

Proof

Step Hyp Ref Expression
1 brfvidRP.r ( 𝜑𝑅 ∈ V )
2 dfid6 I = ( 𝑟 ∈ V ↦ 𝑛 ∈ { 1 } ( 𝑟𝑟 𝑛 ) )
3 1nn0 1 ∈ ℕ0
4 snssi ( 1 ∈ ℕ0 → { 1 } ⊆ ℕ0 )
5 3 4 mp1i ( 𝜑 → { 1 } ⊆ ℕ0 )
6 2 1 5 brmptiunrelexpd ( 𝜑 → ( 𝐴 ( I ‘ 𝑅 ) 𝐵 ↔ ∃ 𝑛 ∈ { 1 } 𝐴 ( 𝑅𝑟 𝑛 ) 𝐵 ) )
7 oveq2 ( 𝑛 = 1 → ( 𝑅𝑟 𝑛 ) = ( 𝑅𝑟 1 ) )
8 7 breqd ( 𝑛 = 1 → ( 𝐴 ( 𝑅𝑟 𝑛 ) 𝐵𝐴 ( 𝑅𝑟 1 ) 𝐵 ) )
9 8 rexsng ( 1 ∈ ℕ0 → ( ∃ 𝑛 ∈ { 1 } 𝐴 ( 𝑅𝑟 𝑛 ) 𝐵𝐴 ( 𝑅𝑟 1 ) 𝐵 ) )
10 3 9 mp1i ( 𝜑 → ( ∃ 𝑛 ∈ { 1 } 𝐴 ( 𝑅𝑟 𝑛 ) 𝐵𝐴 ( 𝑅𝑟 1 ) 𝐵 ) )
11 1 relexp1d ( 𝜑 → ( 𝑅𝑟 1 ) = 𝑅 )
12 11 breqd ( 𝜑 → ( 𝐴 ( 𝑅𝑟 1 ) 𝐵𝐴 𝑅 𝐵 ) )
13 6 10 12 3bitrd ( 𝜑 → ( 𝐴 ( I ‘ 𝑅 ) 𝐵𝐴 𝑅 𝐵 ) )