Step | Hyp | Ref
| Expression |
1 | | tglngval.g |
. . . 4
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
2 | | tglngval.p |
. . . . 5
⊢ 𝑃 = (Base‘𝐺) |
3 | | tglngval.l |
. . . . 5
⊢ 𝐿 = (LineG‘𝐺) |
4 | | tglngval.i |
. . . . 5
⊢ 𝐼 = (Itv‘𝐺) |
5 | 2, 3, 4 | tglng 25241 |
. . . 4
⊢ (𝐺 ∈ TarskiG → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})) |
6 | 1, 5 | syl 17 |
. . 3
⊢ (𝜑 → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})) |
7 | 6 | oveqd 6566 |
. 2
⊢ (𝜑 → (𝑋𝐿𝑌) = (𝑋(𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌)) |
8 | | tglngval.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝑃) |
9 | | tglngval.y |
. . . 4
⊢ (𝜑 → 𝑌 ∈ 𝑃) |
10 | | tglngval.z |
. . . . 5
⊢ (𝜑 → 𝑋 ≠ 𝑌) |
11 | 10 | necomd 2837 |
. . . 4
⊢ (𝜑 → 𝑌 ≠ 𝑋) |
12 | | eldifsn 4260 |
. . . 4
⊢ (𝑌 ∈ (𝑃 ∖ {𝑋}) ↔ (𝑌 ∈ 𝑃 ∧ 𝑌 ≠ 𝑋)) |
13 | 9, 11, 12 | sylanbrc 695 |
. . 3
⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋})) |
14 | | fvex 6113 |
. . . . . 6
⊢
(Base‘𝐺)
∈ V |
15 | 2, 14 | eqeltri 2684 |
. . . . 5
⊢ 𝑃 ∈ V |
16 | 15 | rabex 4740 |
. . . 4
⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V |
17 | 16 | a1i 11 |
. . 3
⊢ (𝜑 → {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V) |
18 | | oveq12 6558 |
. . . . . . 7
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐼𝑦) = (𝑋𝐼𝑌)) |
19 | 18 | eleq2d 2673 |
. . . . . 6
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌))) |
20 | | simpl 472 |
. . . . . . 7
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) |
21 | | simpr 476 |
. . . . . . . 8
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) |
22 | 21 | oveq2d 6565 |
. . . . . . 7
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧𝐼𝑦) = (𝑧𝐼𝑌)) |
23 | 20, 22 | eleq12d 2682 |
. . . . . 6
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌))) |
24 | 20 | oveq1d 6564 |
. . . . . . 7
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐼𝑧) = (𝑋𝐼𝑧)) |
25 | 21, 24 | eleq12d 2682 |
. . . . . 6
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧))) |
26 | 19, 23, 25 | 3orbi123d 1390 |
. . . . 5
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)))) |
27 | 26 | rabbidv 3164 |
. . . 4
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))}) |
28 | | sneq 4135 |
. . . . 5
⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) |
29 | 28 | difeq2d 3690 |
. . . 4
⊢ (𝑥 = 𝑋 → (𝑃 ∖ {𝑥}) = (𝑃 ∖ {𝑋})) |
30 | | eqid 2610 |
. . . 4
⊢ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) |
31 | 27, 29, 30 | ovmpt2x 6687 |
. . 3
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ (𝑃 ∖ {𝑋}) ∧ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V) → (𝑋(𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))}) |
32 | 8, 13, 17, 31 | syl3anc 1318 |
. 2
⊢ (𝜑 → (𝑋(𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))}) |
33 | 7, 32 | eqtrd 2644 |
1
⊢ (𝜑 → (𝑋𝐿𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))}) |