| Step | Hyp | Ref
| Expression |
| 1 | | simpl 482 |
. . . . . 6
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴) |
| 2 | 1 | neeq1d 3000 |
. . . . 5
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝐶 ↔ 𝐴 ≠ 𝐶)) |
| 3 | | simpr 484 |
. . . . . 6
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵) |
| 4 | 3 | neeq1d 3000 |
. . . . 5
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑏 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) |
| 5 | 3 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝐶𝐼𝑏) = (𝐶𝐼𝐵)) |
| 6 | 1, 5 | eleq12d 2835 |
. . . . . 6
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ∈ (𝐶𝐼𝑏) ↔ 𝐴 ∈ (𝐶𝐼𝐵))) |
| 7 | 1 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝐶𝐼𝑎) = (𝐶𝐼𝐴)) |
| 8 | 3, 7 | eleq12d 2835 |
. . . . . 6
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑏 ∈ (𝐶𝐼𝑎) ↔ 𝐵 ∈ (𝐶𝐼𝐴))) |
| 9 | 6, 8 | orbi12d 919 |
. . . . 5
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)) ↔ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))) |
| 10 | 2, 4, 9 | 3anbi123d 1438 |
. . . 4
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))) ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
| 11 | | eqid 2737 |
. . . 4
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} |
| 12 | 10, 11 | brab2a 5779 |
. . 3
⊢ (𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵 ↔ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
| 13 | 12 | a1i 11 |
. 2
⊢ (𝜑 → (𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵 ↔ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))) |
| 14 | | ishlg.k |
. . . . 5
⊢ 𝐾 = (hlG‘𝐺) |
| 15 | | ishlg.g |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ 𝑉) |
| 16 | | elex 3501 |
. . . . . 6
⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) |
| 17 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) |
| 18 | | ishlg.p |
. . . . . . . . 9
⊢ 𝑃 = (Base‘𝐺) |
| 19 | 17, 18 | eqtr4di 2795 |
. . . . . . . 8
⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃) |
| 20 | 19 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔) ↔ 𝑎 ∈ 𝑃)) |
| 21 | 19 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝐺 → (𝑏 ∈ (Base‘𝑔) ↔ 𝑏 ∈ 𝑃)) |
| 22 | 20, 21 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑔 = 𝐺 → ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ↔ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃))) |
| 23 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺)) |
| 24 | | ishlg.i |
. . . . . . . . . . . . . . 15
⊢ 𝐼 = (Itv‘𝐺) |
| 25 | 23, 24 | eqtr4di 2795 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼) |
| 26 | 25 | oveqd 7448 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑏) = (𝑐𝐼𝑏)) |
| 27 | 26 | eleq2d 2827 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝐺 → (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ↔ 𝑎 ∈ (𝑐𝐼𝑏))) |
| 28 | 25 | oveqd 7448 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑎) = (𝑐𝐼𝑎)) |
| 29 | 28 | eleq2d 2827 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝐺 → (𝑏 ∈ (𝑐(Itv‘𝑔)𝑎) ↔ 𝑏 ∈ (𝑐𝐼𝑎))) |
| 30 | 27, 29 | orbi12d 919 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝐺 → ((𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)) ↔ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))) |
| 31 | 30 | 3anbi3d 1444 |
. . . . . . . . . 10
⊢ (𝑔 = 𝐺 → ((𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))) ↔ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))) |
| 32 | 22, 31 | anbi12d 632 |
. . . . . . . . 9
⊢ (𝑔 = 𝐺 → (((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)))) ↔ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))))) |
| 33 | 32 | opabbidv 5209 |
. . . . . . . 8
⊢ (𝑔 = 𝐺 → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}) |
| 34 | 19, 33 | mpteq12dv 5233 |
. . . . . . 7
⊢ (𝑔 = 𝐺 → (𝑐 ∈ (Base‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}) = (𝑐 ∈ 𝑃 ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))})) |
| 35 | | df-hlg 28609 |
. . . . . . 7
⊢ hlG =
(𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))})) |
| 36 | 34, 35, 18 | mptfvmpt 7248 |
. . . . . 6
⊢ (𝐺 ∈ V →
(hlG‘𝐺) = (𝑐 ∈ 𝑃 ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))})) |
| 37 | 15, 16, 36 | 3syl 18 |
. . . . 5
⊢ (𝜑 → (hlG‘𝐺) = (𝑐 ∈ 𝑃 ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))})) |
| 38 | 14, 37 | eqtrid 2789 |
. . . 4
⊢ (𝜑 → 𝐾 = (𝑐 ∈ 𝑃 ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))})) |
| 39 | | neeq2 3004 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → (𝑎 ≠ 𝑐 ↔ 𝑎 ≠ 𝐶)) |
| 40 | | neeq2 3004 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → (𝑏 ≠ 𝑐 ↔ 𝑏 ≠ 𝐶)) |
| 41 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐶 → (𝑐𝐼𝑏) = (𝐶𝐼𝑏)) |
| 42 | 41 | eleq2d 2827 |
. . . . . . . . 9
⊢ (𝑐 = 𝐶 → (𝑎 ∈ (𝑐𝐼𝑏) ↔ 𝑎 ∈ (𝐶𝐼𝑏))) |
| 43 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐶 → (𝑐𝐼𝑎) = (𝐶𝐼𝑎)) |
| 44 | 43 | eleq2d 2827 |
. . . . . . . . 9
⊢ (𝑐 = 𝐶 → (𝑏 ∈ (𝑐𝐼𝑎) ↔ 𝑏 ∈ (𝐶𝐼𝑎))) |
| 45 | 42, 44 | orbi12d 919 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → ((𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)) ↔ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)))) |
| 46 | 39, 40, 45 | 3anbi123d 1438 |
. . . . . . 7
⊢ (𝑐 = 𝐶 → ((𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))) ↔ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))) |
| 47 | 46 | anbi2d 630 |
. . . . . 6
⊢ (𝑐 = 𝐶 → (((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))) ↔ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)))))) |
| 48 | 47 | opabbidv 5209 |
. . . . 5
⊢ (𝑐 = 𝐶 → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}) |
| 49 | 48 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 = 𝐶) → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}) |
| 50 | | ishlg.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
| 51 | 18 | fvexi 6920 |
. . . . . . 7
⊢ 𝑃 ∈ V |
| 52 | 51, 51 | xpex 7773 |
. . . . . 6
⊢ (𝑃 × 𝑃) ∈ V |
| 53 | | opabssxp 5778 |
. . . . . 6
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ⊆ (𝑃 × 𝑃) |
| 54 | 52, 53 | ssexi 5322 |
. . . . 5
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ∈ V |
| 55 | 54 | a1i 11 |
. . . 4
⊢ (𝜑 → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ∈ V) |
| 56 | 38, 49, 50, 55 | fvmptd 7023 |
. . 3
⊢ (𝜑 → (𝐾‘𝐶) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}) |
| 57 | 56 | breqd 5154 |
. 2
⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ 𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵)) |
| 58 | | ishlg.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| 59 | | ishlg.b |
. . . 4
⊢ (𝜑 → 𝐵 ∈ 𝑃) |
| 60 | 58, 59 | jca 511 |
. . 3
⊢ (𝜑 → (𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃)) |
| 61 | 60 | biantrurd 532 |
. 2
⊢ (𝜑 → ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) ↔ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))) |
| 62 | 13, 57, 61 | 3bitr4d 311 |
1
⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |