Step | Hyp | Ref
| Expression |
1 | | sprsymrelf.p |
. . 3
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) |
2 | | sprsymrelf.r |
. . 3
⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} |
3 | | sprsymrelf.f |
. . 3
⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) |
4 | 1, 2, 3 | sprsymrelf 44620 |
. 2
⊢ 𝐹:𝑃⟶𝑅 |
5 | 1, 2, 3 | sprsymrelfv 44619 |
. . . . 5
⊢ (𝑎 ∈ 𝑃 → (𝐹‘𝑎) = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}}) |
6 | 1, 2, 3 | sprsymrelfv 44619 |
. . . . 5
⊢ (𝑏 ∈ 𝑃 → (𝐹‘𝑏) = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) |
7 | 5, 6 | eqeqan12d 2751 |
. . . 4
⊢ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) |
8 | 1 | eleq2i 2829 |
. . . . . 6
⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ∈ 𝒫 (Pairs‘𝑉)) |
9 | | vex 3412 |
. . . . . . 7
⊢ 𝑎 ∈ V |
10 | 9 | elpw 4517 |
. . . . . 6
⊢ (𝑎 ∈ 𝒫
(Pairs‘𝑉) ↔
𝑎 ⊆
(Pairs‘𝑉)) |
11 | 8, 10 | bitri 278 |
. . . . 5
⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ⊆ (Pairs‘𝑉)) |
12 | 1 | eleq2i 2829 |
. . . . . 6
⊢ (𝑏 ∈ 𝑃 ↔ 𝑏 ∈ 𝒫 (Pairs‘𝑉)) |
13 | | vex 3412 |
. . . . . . 7
⊢ 𝑏 ∈ V |
14 | 13 | elpw 4517 |
. . . . . 6
⊢ (𝑏 ∈ 𝒫
(Pairs‘𝑉) ↔
𝑏 ⊆
(Pairs‘𝑉)) |
15 | 12, 14 | bitri 278 |
. . . . 5
⊢ (𝑏 ∈ 𝑃 ↔ 𝑏 ⊆ (Pairs‘𝑉)) |
16 | | sprsymrelf1lem 44616 |
. . . . . . . 8
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 ⊆ 𝑏)) |
17 | 16 | imp 410 |
. . . . . . 7
⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎 ⊆ 𝑏) |
18 | | eqcom 2744 |
. . . . . . . . . 10
⊢
({〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}}) |
19 | | sprsymrelf1lem 44616 |
. . . . . . . . . 10
⊢ ((𝑏 ⊆ (Pairs‘𝑉) ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ({〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} → 𝑏 ⊆ 𝑎)) |
20 | 18, 19 | syl5bi 245 |
. . . . . . . . 9
⊢ ((𝑏 ⊆ (Pairs‘𝑉) ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ({〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑏 ⊆ 𝑎)) |
21 | 20 | ancoms 462 |
. . . . . . . 8
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑏 ⊆ 𝑎)) |
22 | 21 | imp 410 |
. . . . . . 7
⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑏 ⊆ 𝑎) |
23 | 17, 22 | eqssd 3918 |
. . . . . 6
⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎 = 𝑏) |
24 | 23 | ex 416 |
. . . . 5
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 = 𝑏)) |
25 | 11, 15, 24 | syl2anb 601 |
. . . 4
⊢ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) → ({〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 = 𝑏)) |
26 | 7, 25 | sylbid 243 |
. . 3
⊢ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)) |
27 | 26 | rgen2 3124 |
. 2
⊢
∀𝑎 ∈
𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏) |
28 | | dff13 7067 |
. 2
⊢ (𝐹:𝑃–1-1→𝑅 ↔ (𝐹:𝑃⟶𝑅 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))) |
29 | 4, 27, 28 | mpbir2an 711 |
1
⊢ 𝐹:𝑃–1-1→𝑅 |