| Step | Hyp | Ref
| Expression |
| 1 | | utopustuq.1 |
. . . . 5
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) |
| 2 | | simpl 482 |
. . . . . . . . . 10
⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → 𝑝 = 𝑞) |
| 3 | 2 | sneqd 4638 |
. . . . . . . . 9
⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → {𝑝} = {𝑞}) |
| 4 | 3 | imaeq2d 6078 |
. . . . . . . 8
⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑞})) |
| 5 | 4 | mpteq2dva 5242 |
. . . . . . 7
⊢ (𝑝 = 𝑞 → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) |
| 6 | 5 | rneqd 5949 |
. . . . . 6
⊢ (𝑝 = 𝑞 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) |
| 7 | 6 | cbvmptv 5255 |
. . . . 5
⊢ (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) = (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) |
| 8 | 1, 7 | eqtri 2765 |
. . . 4
⊢ 𝑁 = (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) |
| 9 | | simpr2 1196 |
. . . . . . . . 9
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → 𝑞 = 𝑃) |
| 10 | 9 | sneqd 4638 |
. . . . . . . 8
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → {𝑞} = {𝑃}) |
| 11 | 10 | imaeq2d 6078 |
. . . . . . 7
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃})) |
| 12 | 11 | 3anassrs 1361 |
. . . . . 6
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃})) |
| 13 | 12 | mpteq2dva 5242 |
. . . . 5
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
| 14 | 13 | rneqd 5949 |
. . . 4
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
| 15 | | simpr 484 |
. . . 4
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋) |
| 16 | | mptexg 7241 |
. . . . . 6
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) |
| 17 | | rnexg 7924 |
. . . . . 6
⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) |
| 18 | 16, 17 | syl 17 |
. . . . 5
⊢ (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) |
| 19 | 18 | adantr 480 |
. . . 4
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) |
| 20 | 8, 14, 15, 19 | fvmptd2 7024 |
. . 3
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
| 21 | 20 | eleq2d 2827 |
. 2
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑁‘𝑃) ↔ 𝐴 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))) |
| 22 | | imaeq1 6073 |
. . . 4
⊢ (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃})) |
| 23 | 22 | cbvmptv 5255 |
. . 3
⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑤 ∈ 𝑈 ↦ (𝑤 “ {𝑃})) |
| 24 | 23 | elrnmpt 5969 |
. 2
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃}))) |
| 25 | 21, 24 | sylan9bb 509 |
1
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑁‘𝑃) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃}))) |