Proof of Theorem f1o00
Step | Hyp | Ref
| Expression |
1 | | dff1o4 6838 |
. 2
⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴)) |
2 | | fn0 6678 |
. . . . . 6
⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) |
3 | 2 | biimpi 215 |
. . . . 5
⊢ (𝐹 Fn ∅ → 𝐹 = ∅) |
4 | 3 | adantr 482 |
. . . 4
⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐹 = ∅) |
5 | | cnveq 5871 |
. . . . . . . . . 10
⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅) |
6 | | cnv0 6137 |
. . . . . . . . . 10
⊢ ◡∅ = ∅ |
7 | 5, 6 | eqtrdi 2789 |
. . . . . . . . 9
⊢ (𝐹 = ∅ → ◡𝐹 = ∅) |
8 | 2, 7 | sylbi 216 |
. . . . . . . 8
⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅) |
9 | 8 | fneq1d 6639 |
. . . . . . 7
⊢ (𝐹 Fn ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴)) |
10 | 9 | biimpa 478 |
. . . . . 6
⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → ∅ Fn 𝐴) |
11 | 10 | fndmd 6651 |
. . . . 5
⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → dom ∅ = 𝐴) |
12 | | dm0 5918 |
. . . . 5
⊢ dom
∅ = ∅ |
13 | 11, 12 | eqtr3di 2788 |
. . . 4
⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐴 = ∅) |
14 | 4, 13 | jca 513 |
. . 3
⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅)) |
15 | 2 | biimpri 227 |
. . . . 5
⊢ (𝐹 = ∅ → 𝐹 Fn ∅) |
16 | 15 | adantr 482 |
. . . 4
⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅) |
17 | | eqid 2733 |
. . . . . 6
⊢ ∅ =
∅ |
18 | | fn0 6678 |
. . . . . 6
⊢ (∅
Fn ∅ ↔ ∅ = ∅) |
19 | 17, 18 | mpbir 230 |
. . . . 5
⊢ ∅
Fn ∅ |
20 | 7 | fneq1d 6639 |
. . . . . 6
⊢ (𝐹 = ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴)) |
21 | | fneq2 6638 |
. . . . . 6
⊢ (𝐴 = ∅ → (∅ Fn
𝐴 ↔ ∅ Fn
∅)) |
22 | 20, 21 | sylan9bb 511 |
. . . . 5
⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (◡𝐹 Fn 𝐴 ↔ ∅ Fn
∅)) |
23 | 19, 22 | mpbiri 258 |
. . . 4
⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → ◡𝐹 Fn 𝐴) |
24 | 16, 23 | jca 513 |
. . 3
⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴)) |
25 | 14, 24 | impbii 208 |
. 2
⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
26 | 1, 25 | bitri 275 |
1
⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |