Step |
Hyp |
Ref |
Expression |
1 |
|
releldm2 |
⊢ ( Rel 𝐴 → ( 𝑦 ∈ dom 𝐴 ↔ ∃ 𝑧 ∈ 𝐴 ( 1st ‘ 𝑧 ) = 𝑦 ) ) |
2 |
|
fvex |
⊢ ( 1st ‘ 𝑥 ) ∈ V |
3 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) |
4 |
2 3
|
fnmpti |
⊢ ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) Fn 𝐴 |
5 |
|
fvelrnb |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) Fn 𝐴 → ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ↔ ∃ 𝑧 ∈ 𝐴 ( ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ‘ 𝑧 ) = 𝑦 ) ) |
6 |
4 5
|
ax-mp |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ↔ ∃ 𝑧 ∈ 𝐴 ( ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ‘ 𝑧 ) = 𝑦 ) |
7 |
|
fveq2 |
⊢ ( 𝑥 = 𝑧 → ( 1st ‘ 𝑥 ) = ( 1st ‘ 𝑧 ) ) |
8 |
|
fvex |
⊢ ( 1st ‘ 𝑧 ) ∈ V |
9 |
7 3 8
|
fvmpt |
⊢ ( 𝑧 ∈ 𝐴 → ( ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ‘ 𝑧 ) = ( 1st ‘ 𝑧 ) ) |
10 |
9
|
eqeq1d |
⊢ ( 𝑧 ∈ 𝐴 → ( ( ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ‘ 𝑧 ) = 𝑦 ↔ ( 1st ‘ 𝑧 ) = 𝑦 ) ) |
11 |
10
|
rexbiia |
⊢ ( ∃ 𝑧 ∈ 𝐴 ( ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ‘ 𝑧 ) = 𝑦 ↔ ∃ 𝑧 ∈ 𝐴 ( 1st ‘ 𝑧 ) = 𝑦 ) |
12 |
11
|
a1i |
⊢ ( Rel 𝐴 → ( ∃ 𝑧 ∈ 𝐴 ( ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ‘ 𝑧 ) = 𝑦 ↔ ∃ 𝑧 ∈ 𝐴 ( 1st ‘ 𝑧 ) = 𝑦 ) ) |
13 |
6 12
|
bitr2id |
⊢ ( Rel 𝐴 → ( ∃ 𝑧 ∈ 𝐴 ( 1st ‘ 𝑧 ) = 𝑦 ↔ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ) ) |
14 |
1 13
|
bitrd |
⊢ ( Rel 𝐴 → ( 𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ) ) |
15 |
14
|
eqrdv |
⊢ ( Rel 𝐴 → dom 𝐴 = ran ( 𝑥 ∈ 𝐴 ↦ ( 1st ‘ 𝑥 ) ) ) |