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 𝑦 ↔ 𝑦 = ( 𝐴 “ 𝑥 ) ) |
10 |
|
vex |
⊢ 𝑧 ∈ V |
11 |
7 10
|
brimage |
⊢ ( 𝑥 Image 𝐴 𝑧 ↔ 𝑧 = ( 𝐴 “ 𝑥 ) ) |
12 |
|
eqtr3 |
⊢ ( ( 𝑦 = ( 𝐴 “ 𝑥 ) ∧ 𝑧 = ( 𝐴 “ 𝑥 ) ) → 𝑦 = 𝑧 ) |
13 |
9 11 12
|
syl2anb |
⊢ ( ( 𝑥 Image 𝐴 𝑦 ∧ 𝑥 Image 𝐴 𝑧 ) → 𝑦 = 𝑧 ) |
14 |
13
|
gen2 |
⊢ ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 Image 𝐴 𝑦 ∧ 𝑥 Image 𝐴 𝑧 ) → 𝑦 = 𝑧 ) |
15 |
14
|
ax-gen |
⊢ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 Image 𝐴 𝑦 ∧ 𝑥 Image 𝐴 𝑧 ) → 𝑦 = 𝑧 ) |
16 |
|
dffun2 |
⊢ ( Fun Image 𝐴 ↔ ( Rel Image 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 Image 𝐴 𝑦 ∧ 𝑥 Image 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
17 |
6 15 16
|
mpbir2an |
⊢ Fun Image 𝐴 |