Proof of Theorem f1o3d
Step | Hyp | Ref
| Expression |
1 | | f1o3d.2 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
2 | 1 | ralrimiva 3140 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) |
3 | | eqid 2737 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
4 | 3 | fnmpt 6610 |
. . . . 5
⊢
(∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) |
5 | 2, 4 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) |
6 | | f1o3d.1 |
. . . . 5
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
7 | 6 | fneq1d 6564 |
. . . 4
⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)) |
8 | 5, 7 | mpbird 256 |
. . 3
⊢ (𝜑 → 𝐹 Fn 𝐴) |
9 | | f1o3d.3 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) |
10 | 9 | ralrimiva 3140 |
. . . . 5
⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝐷 ∈ 𝐴) |
11 | | eqid 2737 |
. . . . . 6
⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = (𝑦 ∈ 𝐵 ↦ 𝐷) |
12 | 11 | fnmpt 6610 |
. . . . 5
⊢
(∀𝑦 ∈
𝐵 𝐷 ∈ 𝐴 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) |
13 | 10, 12 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) |
14 | | eleq1a 2833 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵)) |
15 | 1, 14 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵)) |
16 | 15 | impr 455 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → 𝑦 ∈ 𝐵) |
17 | | f1o3d.4 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) |
18 | 17 | biimpar 478 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷) |
19 | 18 | exp42 436 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑥 = 𝐷)))) |
20 | 19 | com34 91 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷)))) |
21 | 20 | imp32 419 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷)) |
22 | 16, 21 | jcai 517 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) |
23 | | eleq1a 2833 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴)) |
24 | 9, 23 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴)) |
25 | 24 | impr 455 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → 𝑥 ∈ 𝐴) |
26 | 17 | biimpa 477 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶) |
27 | 26 | exp42 436 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → 𝑦 = 𝐶)))) |
28 | 27 | com23 86 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑦 = 𝐶)))) |
29 | 28 | com34 91 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶)))) |
30 | 29 | imp32 419 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶)) |
31 | 25, 30 | jcai 517 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) |
32 | 22, 31 | impbida 798 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) |
33 | 32 | opabbidv 5153 |
. . . . . 6
⊢ (𝜑 → {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)}) |
34 | | df-mpt 5171 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
35 | 6, 34 | eqtrdi 2793 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}) |
36 | 35 | cnveqd 5804 |
. . . . . . 7
⊢ (𝜑 → ◡𝐹 = ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}) |
37 | | cnvopab 6064 |
. . . . . . 7
⊢ ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
38 | 36, 37 | eqtrdi 2793 |
. . . . . 6
⊢ (𝜑 → ◡𝐹 = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}) |
39 | | df-mpt 5171 |
. . . . . . 7
⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)} |
40 | 39 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)}) |
41 | 33, 38, 40 | 3eqtr4d 2787 |
. . . . 5
⊢ (𝜑 → ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)) |
42 | 41 | fneq1d 6564 |
. . . 4
⊢ (𝜑 → (◡𝐹 Fn 𝐵 ↔ (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵)) |
43 | 13, 42 | mpbird 256 |
. . 3
⊢ (𝜑 → ◡𝐹 Fn 𝐵) |
44 | | dff1o4 6761 |
. . 3
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) |
45 | 8, 43, 44 | sylanbrc 583 |
. 2
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
46 | 45, 41 | jca 512 |
1
⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |