Metamath Proof Explorer


Theorem fnimage

Description: Image R is a function over the set-like portion of R . (Contributed by Scott Fenton, 4-Apr-2014) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Assertion fnimage Image 𝑅 Fn { 𝑥 ∣ ( 𝑅𝑥 ) ∈ V }

Proof

Step Hyp Ref Expression
1 funimage Fun Image 𝑅
2 vex 𝑦 ∈ V
3 vex 𝑥 ∈ V
4 2 3 brimage ( 𝑦 Image 𝑅 𝑥𝑥 = ( 𝑅𝑦 ) )
5 eqvisset ( 𝑥 = ( 𝑅𝑦 ) → ( 𝑅𝑦 ) ∈ V )
6 4 5 sylbi ( 𝑦 Image 𝑅 𝑥 → ( 𝑅𝑦 ) ∈ V )
7 6 exlimiv ( ∃ 𝑥 𝑦 Image 𝑅 𝑥 → ( 𝑅𝑦 ) ∈ V )
8 eqid ( 𝑅𝑦 ) = ( 𝑅𝑦 )
9 brimageg ( ( 𝑦 ∈ V ∧ ( 𝑅𝑦 ) ∈ V ) → ( 𝑦 Image 𝑅 ( 𝑅𝑦 ) ↔ ( 𝑅𝑦 ) = ( 𝑅𝑦 ) ) )
10 2 9 mpan ( ( 𝑅𝑦 ) ∈ V → ( 𝑦 Image 𝑅 ( 𝑅𝑦 ) ↔ ( 𝑅𝑦 ) = ( 𝑅𝑦 ) ) )
11 8 10 mpbiri ( ( 𝑅𝑦 ) ∈ V → 𝑦 Image 𝑅 ( 𝑅𝑦 ) )
12 breq2 ( 𝑥 = ( 𝑅𝑦 ) → ( 𝑦 Image 𝑅 𝑥𝑦 Image 𝑅 ( 𝑅𝑦 ) ) )
13 12 spcegv ( ( 𝑅𝑦 ) ∈ V → ( 𝑦 Image 𝑅 ( 𝑅𝑦 ) → ∃ 𝑥 𝑦 Image 𝑅 𝑥 ) )
14 11 13 mpd ( ( 𝑅𝑦 ) ∈ V → ∃ 𝑥 𝑦 Image 𝑅 𝑥 )
15 7 14 impbii ( ∃ 𝑥 𝑦 Image 𝑅 𝑥 ↔ ( 𝑅𝑦 ) ∈ V )
16 2 eldm ( 𝑦 ∈ dom Image 𝑅 ↔ ∃ 𝑥 𝑦 Image 𝑅 𝑥 )
17 imaeq2 ( 𝑥 = 𝑦 → ( 𝑅𝑥 ) = ( 𝑅𝑦 ) )
18 17 eleq1d ( 𝑥 = 𝑦 → ( ( 𝑅𝑥 ) ∈ V ↔ ( 𝑅𝑦 ) ∈ V ) )
19 2 18 elab ( 𝑦 ∈ { 𝑥 ∣ ( 𝑅𝑥 ) ∈ V } ↔ ( 𝑅𝑦 ) ∈ V )
20 15 16 19 3bitr4i ( 𝑦 ∈ dom Image 𝑅𝑦 ∈ { 𝑥 ∣ ( 𝑅𝑥 ) ∈ V } )
21 20 eqriv dom Image 𝑅 = { 𝑥 ∣ ( 𝑅𝑥 ) ∈ V }
22 df-fn ( Image 𝑅 Fn { 𝑥 ∣ ( 𝑅𝑥 ) ∈ V } ↔ ( Fun Image 𝑅 ∧ dom Image 𝑅 = { 𝑥 ∣ ( 𝑅𝑥 ) ∈ V } ) )
23 1 21 22 mpbir2an Image 𝑅 Fn { 𝑥 ∣ ( 𝑅𝑥 ) ∈ V }