Step |
Hyp |
Ref |
Expression |
1 |
|
pjmfn |
⊢ projℎ Fn Cℋ |
2 |
|
pjhf |
⊢ ( 𝑥 ∈ Cℋ → ( projℎ ‘ 𝑥 ) : ℋ ⟶ ℋ ) |
3 |
|
ax-hilex |
⊢ ℋ ∈ V |
4 |
3 3
|
elmap |
⊢ ( ( projℎ ‘ 𝑥 ) ∈ ( ℋ ↑m ℋ ) ↔ ( projℎ ‘ 𝑥 ) : ℋ ⟶ ℋ ) |
5 |
2 4
|
sylibr |
⊢ ( 𝑥 ∈ Cℋ → ( projℎ ‘ 𝑥 ) ∈ ( ℋ ↑m ℋ ) ) |
6 |
5
|
rgen |
⊢ ∀ 𝑥 ∈ Cℋ ( projℎ ‘ 𝑥 ) ∈ ( ℋ ↑m ℋ ) |
7 |
|
ffnfv |
⊢ ( projℎ : Cℋ ⟶ ( ℋ ↑m ℋ ) ↔ ( projℎ Fn Cℋ ∧ ∀ 𝑥 ∈ Cℋ ( projℎ ‘ 𝑥 ) ∈ ( ℋ ↑m ℋ ) ) ) |
8 |
1 6 7
|
mpbir2an |
⊢ projℎ : Cℋ ⟶ ( ℋ ↑m ℋ ) |
9 |
|
pj11 |
⊢ ( ( 𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) → ( ( projℎ ‘ 𝑥 ) = ( projℎ ‘ 𝑦 ) ↔ 𝑥 = 𝑦 ) ) |
10 |
9
|
biimpd |
⊢ ( ( 𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) → ( ( projℎ ‘ 𝑥 ) = ( projℎ ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
11 |
10
|
rgen2 |
⊢ ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( ( projℎ ‘ 𝑥 ) = ( projℎ ‘ 𝑦 ) → 𝑥 = 𝑦 ) |
12 |
|
dff13 |
⊢ ( projℎ : Cℋ –1-1→ ( ℋ ↑m ℋ ) ↔ ( projℎ : Cℋ ⟶ ( ℋ ↑m ℋ ) ∧ ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( ( projℎ ‘ 𝑥 ) = ( projℎ ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
13 |
8 11 12
|
mpbir2an |
⊢ projℎ : Cℋ –1-1→ ( ℋ ↑m ℋ ) |