Metamath Proof Explorer


Theorem funimage

Description: Image A is a function. (Contributed by Scott Fenton, 27-Mar-2014) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Assertion funimage Fun Image 𝐴

Proof

Step Hyp Ref Expression
1 difss ( ( V × V ) ∖ ran ( ( V ⊗ E ) △ ( ( E ∘ 𝐴 ) ⊗ V ) ) ) ⊆ ( V × V )
2 df-rel ( Rel ( ( V × V ) ∖ ran ( ( V ⊗ E ) △ ( ( E ∘ 𝐴 ) ⊗ V ) ) ) ↔ ( ( V × V ) ∖ ran ( ( V ⊗ E ) △ ( ( E ∘ 𝐴 ) ⊗ V ) ) ) ⊆ ( V × V ) )
3 1 2 mpbir Rel ( ( V × V ) ∖ ran ( ( V ⊗ E ) △ ( ( E ∘ 𝐴 ) ⊗ V ) ) )
4 df-image Image 𝐴 = ( ( V × V ) ∖ ran ( ( V ⊗ E ) △ ( ( E ∘ 𝐴 ) ⊗ V ) ) )
5 4 releqi ( Rel Image 𝐴 ↔ Rel ( ( V × V ) ∖ ran ( ( V ⊗ E ) △ ( ( E ∘ 𝐴 ) ⊗ V ) ) ) )
6 3 5 mpbir Rel Image 𝐴
7 vex 𝑥 ∈ V
8 vex 𝑦 ∈ V
9 7 8 brimage ( 𝑥 Image 𝐴 𝑦𝑦 = ( 𝐴𝑥 ) )
10 vex 𝑧 ∈ V
11 7 10 brimage ( 𝑥 Image 𝐴 𝑧𝑧 = ( 𝐴𝑥 ) )
12 eqtr3 ( ( 𝑦 = ( 𝐴𝑥 ) ∧ 𝑧 = ( 𝐴𝑥 ) ) → 𝑦 = 𝑧 )
13 9 11 12 syl2anb ( ( 𝑥 Image 𝐴 𝑦𝑥 Image 𝐴 𝑧 ) → 𝑦 = 𝑧 )
14 13 gen2 𝑦𝑧 ( ( 𝑥 Image 𝐴 𝑦𝑥 Image 𝐴 𝑧 ) → 𝑦 = 𝑧 )
15 14 ax-gen 𝑥𝑦𝑧 ( ( 𝑥 Image 𝐴 𝑦𝑥 Image 𝐴 𝑧 ) → 𝑦 = 𝑧 )
16 dffun2 ( Fun Image 𝐴 ↔ ( Rel Image 𝐴 ∧ ∀ 𝑥𝑦𝑧 ( ( 𝑥 Image 𝐴 𝑦𝑥 Image 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ) )
17 6 15 16 mpbir2an Fun Image 𝐴