| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ishpg.g | . . . 4
⊢ (𝜑 → 𝐺 ∈ TarskiG) | 
| 2 |  | elex 3500 | . . . 4
⊢ (𝐺 ∈ TarskiG → 𝐺 ∈ V) | 
| 3 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺)) | 
| 4 |  | ishpg.l | . . . . . . . 8
⊢ 𝐿 = (LineG‘𝐺) | 
| 5 | 3, 4 | eqtr4di 2794 | . . . . . . 7
⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿) | 
| 6 | 5 | rneqd 5948 | . . . . . 6
⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿) | 
| 7 |  | ishpg.p | . . . . . . . 8
⊢ 𝑃 = (Base‘𝐺) | 
| 8 |  | ishpg.i | . . . . . . . 8
⊢ 𝐼 = (Itv‘𝐺) | 
| 9 |  | simpl 482 | . . . . . . . . 9
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝑝 = 𝑃) | 
| 10 | 9 | difeq1d 4124 | . . . . . . . . . . . . 13
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑝 ∖ 𝑑) = (𝑃 ∖ 𝑑)) | 
| 11 | 10 | eleq2d 2826 | . . . . . . . . . . . 12
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑎 ∈ (𝑝 ∖ 𝑑) ↔ 𝑎 ∈ (𝑃 ∖ 𝑑))) | 
| 12 | 10 | eleq2d 2826 | . . . . . . . . . . . 12
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑐 ∈ (𝑝 ∖ 𝑑) ↔ 𝑐 ∈ (𝑃 ∖ 𝑑))) | 
| 13 | 11, 12 | anbi12d 632 | . . . . . . . . . . 11
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → ((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ↔ (𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)))) | 
| 14 |  | simpr 484 | . . . . . . . . . . . . . 14
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼) | 
| 15 | 14 | oveqd 7449 | . . . . . . . . . . . . 13
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑎𝑖𝑐) = (𝑎𝐼𝑐)) | 
| 16 | 15 | eleq2d 2826 | . . . . . . . . . . . 12
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑡 ∈ (𝑎𝑖𝑐) ↔ 𝑡 ∈ (𝑎𝐼𝑐))) | 
| 17 | 16 | rexbidv 3178 | . . . . . . . . . . 11
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐) ↔ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐))) | 
| 18 | 13, 17 | anbi12d 632 | . . . . . . . . . 10
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ↔ ((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)))) | 
| 19 | 10 | eleq2d 2826 | . . . . . . . . . . . 12
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑏 ∈ (𝑝 ∖ 𝑑) ↔ 𝑏 ∈ (𝑃 ∖ 𝑑))) | 
| 20 | 19, 12 | anbi12d 632 | . . . . . . . . . . 11
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ↔ (𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)))) | 
| 21 | 14 | oveqd 7449 | . . . . . . . . . . . . 13
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑏𝑖𝑐) = (𝑏𝐼𝑐)) | 
| 22 | 21 | eleq2d 2826 | . . . . . . . . . . . 12
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑡 ∈ (𝑏𝑖𝑐) ↔ 𝑡 ∈ (𝑏𝐼𝑐))) | 
| 23 | 22 | rexbidv 3178 | . . . . . . . . . . 11
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐) ↔ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))) | 
| 24 | 20, 23 | anbi12d 632 | . . . . . . . . . 10
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)) ↔ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))) | 
| 25 | 18, 24 | anbi12d 632 | . . . . . . . . 9
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → ((((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐))) ↔ (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))))) | 
| 26 | 9, 25 | rexeqbidv 3346 | . . . . . . . 8
⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐))) ↔ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))))) | 
| 27 | 7, 8, 26 | sbcie2s 17199 | . . . . . . 7
⊢ (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐))) ↔ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))))) | 
| 28 | 27 | opabbidv 5208 | . . . . . 6
⊢ (𝑔 = 𝐺 → {〈𝑎, 𝑏〉 ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))} = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}) | 
| 29 | 6, 28 | mpteq12dv 5232 | . . . . 5
⊢ (𝑔 = 𝐺 → (𝑑 ∈ ran (LineG‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}) = (𝑑 ∈ ran 𝐿 ↦ {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))})) | 
| 30 |  | df-hpg 28767 | . . . . 5
⊢ hpG =
(𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))})) | 
| 31 | 4 | fvexi 6919 | . . . . . . 7
⊢ 𝐿 ∈ V | 
| 32 | 31 | rnex 7933 | . . . . . 6
⊢ ran 𝐿 ∈ V | 
| 33 | 32 | mptex 7244 | . . . . 5
⊢ (𝑑 ∈ ran 𝐿 ↦ {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}) ∈ V | 
| 34 | 29, 30, 33 | fvmpt 7015 | . . . 4
⊢ (𝐺 ∈ V →
(hpG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))})) | 
| 35 | 1, 2, 34 | 3syl 18 | . . 3
⊢ (𝜑 → (hpG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))})) | 
| 36 |  | difeq2 4119 | . . . . . . . . . 10
⊢ (𝑑 = 𝐷 → (𝑃 ∖ 𝑑) = (𝑃 ∖ 𝐷)) | 
| 37 | 36 | eleq2d 2826 | . . . . . . . . 9
⊢ (𝑑 = 𝐷 → (𝑎 ∈ (𝑃 ∖ 𝑑) ↔ 𝑎 ∈ (𝑃 ∖ 𝐷))) | 
| 38 | 36 | eleq2d 2826 | . . . . . . . . 9
⊢ (𝑑 = 𝐷 → (𝑐 ∈ (𝑃 ∖ 𝑑) ↔ 𝑐 ∈ (𝑃 ∖ 𝐷))) | 
| 39 | 37, 38 | anbi12d 632 | . . . . . . . 8
⊢ (𝑑 = 𝐷 → ((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ↔ (𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)))) | 
| 40 |  | rexeq 3321 | . . . . . . . 8
⊢ (𝑑 = 𝐷 → (∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐))) | 
| 41 | 39, 40 | anbi12d 632 | . . . . . . 7
⊢ (𝑑 = 𝐷 → (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ↔ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)))) | 
| 42 | 36 | eleq2d 2826 | . . . . . . . . 9
⊢ (𝑑 = 𝐷 → (𝑏 ∈ (𝑃 ∖ 𝑑) ↔ 𝑏 ∈ (𝑃 ∖ 𝐷))) | 
| 43 | 42, 38 | anbi12d 632 | . . . . . . . 8
⊢ (𝑑 = 𝐷 → ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ↔ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)))) | 
| 44 |  | rexeq 3321 | . . . . . . . 8
⊢ (𝑑 = 𝐷 → (∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))) | 
| 45 | 43, 44 | anbi12d 632 | . . . . . . 7
⊢ (𝑑 = 𝐷 → (((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)) ↔ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))) | 
| 46 | 41, 45 | anbi12d 632 | . . . . . 6
⊢ (𝑑 = 𝐷 → ((((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))))) | 
| 47 | 46 | rexbidv 3178 | . . . . 5
⊢ (𝑑 = 𝐷 → (∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))))) | 
| 48 | 47 | opabbidv 5208 | . . . 4
⊢ (𝑑 = 𝐷 → {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))} = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))}) | 
| 49 | 48 | adantl 481 | . . 3
⊢ ((𝜑 ∧ 𝑑 = 𝐷) → {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))} = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))}) | 
| 50 |  | ishpg.d | . . 3
⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | 
| 51 |  | df-xp 5690 | . . . . . 6
⊢ (𝑃 × 𝑃) = {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)} | 
| 52 | 7 | fvexi 6919 | . . . . . . 7
⊢ 𝑃 ∈ V | 
| 53 | 52, 52 | xpex 7774 | . . . . . 6
⊢ (𝑃 × 𝑃) ∈ V | 
| 54 | 51, 53 | eqeltrri 2837 | . . . . 5
⊢
{〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)} ∈ V | 
| 55 |  | eldifi 4130 | . . . . . . . . . 10
⊢ (𝑎 ∈ (𝑃 ∖ 𝐷) → 𝑎 ∈ 𝑃) | 
| 56 |  | eldifi 4130 | . . . . . . . . . 10
⊢ (𝑏 ∈ (𝑃 ∖ 𝐷) → 𝑏 ∈ 𝑃) | 
| 57 | 55, 56 | anim12i 613 | . . . . . . . . 9
⊢ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) | 
| 58 | 57 | ad2ant2r 747 | . . . . . . . 8
⊢ (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷))) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) | 
| 59 | 58 | ad2ant2r 747 | . . . . . . 7
⊢ ((((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) | 
| 60 | 59 | rexlimivw 3150 | . . . . . 6
⊢
(∃𝑐 ∈
𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) | 
| 61 | 60 | ssopab2i 5554 | . . . . 5
⊢
{〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ⊆ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)} | 
| 62 | 54, 61 | ssexi 5321 | . . . 4
⊢
{〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V | 
| 63 | 62 | a1i 11 | . . 3
⊢ (𝜑 → {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V) | 
| 64 | 35, 49, 50, 63 | fvmptd 7022 | . 2
⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))}) | 
| 65 |  | vex 3483 | . . . . . 6
⊢ 𝑎 ∈ V | 
| 66 |  | vex 3483 | . . . . . 6
⊢ 𝑐 ∈ V | 
| 67 |  | eleq1w 2823 | . . . . . . . 8
⊢ (𝑒 = 𝑎 → (𝑒 ∈ (𝑃 ∖ 𝐷) ↔ 𝑎 ∈ (𝑃 ∖ 𝐷))) | 
| 68 | 67 | anbi1d 631 | . . . . . . 7
⊢ (𝑒 = 𝑎 → ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)))) | 
| 69 |  | oveq1 7439 | . . . . . . . . 9
⊢ (𝑒 = 𝑎 → (𝑒𝐼𝑓) = (𝑎𝐼𝑓)) | 
| 70 | 69 | eleq2d 2826 | . . . . . . . 8
⊢ (𝑒 = 𝑎 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑓))) | 
| 71 | 70 | rexbidv 3178 | . . . . . . 7
⊢ (𝑒 = 𝑎 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑓))) | 
| 72 | 68, 71 | anbi12d 632 | . . . . . 6
⊢ (𝑒 = 𝑎 → (((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑓)))) | 
| 73 |  | eleq1w 2823 | . . . . . . . 8
⊢ (𝑓 = 𝑐 → (𝑓 ∈ (𝑃 ∖ 𝐷) ↔ 𝑐 ∈ (𝑃 ∖ 𝐷))) | 
| 74 | 73 | anbi2d 630 | . . . . . . 7
⊢ (𝑓 = 𝑐 → ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)))) | 
| 75 |  | oveq2 7440 | . . . . . . . . 9
⊢ (𝑓 = 𝑐 → (𝑎𝐼𝑓) = (𝑎𝐼𝑐)) | 
| 76 | 75 | eleq2d 2826 | . . . . . . . 8
⊢ (𝑓 = 𝑐 → (𝑡 ∈ (𝑎𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑐))) | 
| 77 | 76 | rexbidv 3178 | . . . . . . 7
⊢ (𝑓 = 𝑐 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑓) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐))) | 
| 78 | 74, 77 | anbi12d 632 | . . . . . 6
⊢ (𝑓 = 𝑐 → (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)))) | 
| 79 |  | ishpg.o | . . . . . . 7
⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | 
| 80 |  | simpl 482 | . . . . . . . . . . 11
⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → 𝑎 = 𝑒) | 
| 81 | 80 | eleq1d 2825 | . . . . . . . . . 10
⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (𝑎 ∈ (𝑃 ∖ 𝐷) ↔ 𝑒 ∈ (𝑃 ∖ 𝐷))) | 
| 82 |  | simpr 484 | . . . . . . . . . . 11
⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → 𝑏 = 𝑓) | 
| 83 | 82 | eleq1d 2825 | . . . . . . . . . 10
⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (𝑏 ∈ (𝑃 ∖ 𝐷) ↔ 𝑓 ∈ (𝑃 ∖ 𝐷))) | 
| 84 | 81, 83 | anbi12d 632 | . . . . . . . . 9
⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)))) | 
| 85 |  | oveq12 7441 | . . . . . . . . . . 11
⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (𝑎𝐼𝑏) = (𝑒𝐼𝑓)) | 
| 86 | 85 | eleq2d 2826 | . . . . . . . . . 10
⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑒𝐼𝑓))) | 
| 87 | 86 | rexbidv 3178 | . . . . . . . . 9
⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓))) | 
| 88 | 84, 87 | anbi12d 632 | . . . . . . . 8
⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓)))) | 
| 89 | 88 | cbvopabv 5215 | . . . . . . 7
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} = {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓))} | 
| 90 | 79, 89 | eqtri 2764 | . . . . . 6
⊢ 𝑂 = {〈𝑒, 𝑓〉 ∣ ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓))} | 
| 91 | 65, 66, 72, 78, 90 | brab 5547 | . . . . 5
⊢ (𝑎𝑂𝑐 ↔ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐))) | 
| 92 |  | vex 3483 | . . . . . 6
⊢ 𝑏 ∈ V | 
| 93 |  | eleq1w 2823 | . . . . . . . 8
⊢ (𝑒 = 𝑏 → (𝑒 ∈ (𝑃 ∖ 𝐷) ↔ 𝑏 ∈ (𝑃 ∖ 𝐷))) | 
| 94 | 93 | anbi1d 631 | . . . . . . 7
⊢ (𝑒 = 𝑏 → ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)))) | 
| 95 |  | oveq1 7439 | . . . . . . . . 9
⊢ (𝑒 = 𝑏 → (𝑒𝐼𝑓) = (𝑏𝐼𝑓)) | 
| 96 | 95 | eleq2d 2826 | . . . . . . . 8
⊢ (𝑒 = 𝑏 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑓))) | 
| 97 | 96 | rexbidv 3178 | . . . . . . 7
⊢ (𝑒 = 𝑏 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑓))) | 
| 98 | 94, 97 | anbi12d 632 | . . . . . 6
⊢ (𝑒 = 𝑏 → (((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑓)))) | 
| 99 | 73 | anbi2d 630 | . . . . . . 7
⊢ (𝑓 = 𝑐 → ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)))) | 
| 100 |  | oveq2 7440 | . . . . . . . . 9
⊢ (𝑓 = 𝑐 → (𝑏𝐼𝑓) = (𝑏𝐼𝑐)) | 
| 101 | 100 | eleq2d 2826 | . . . . . . . 8
⊢ (𝑓 = 𝑐 → (𝑡 ∈ (𝑏𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑐))) | 
| 102 | 101 | rexbidv 3178 | . . . . . . 7
⊢ (𝑓 = 𝑐 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑓) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))) | 
| 103 | 99, 102 | anbi12d 632 | . . . . . 6
⊢ (𝑓 = 𝑐 → (((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))) | 
| 104 | 92, 66, 98, 103, 90 | brab 5547 | . . . . 5
⊢ (𝑏𝑂𝑐 ↔ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))) | 
| 105 | 91, 104 | anbi12i 628 | . . . 4
⊢ ((𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))) | 
| 106 | 105 | rexbii 3093 | . . 3
⊢
(∃𝑐 ∈
𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))) | 
| 107 | 106 | opabbii 5209 | . 2
⊢
{〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)} = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} | 
| 108 | 64, 107 | eqtr4di 2794 | 1
⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)}) |