Step | Hyp | Ref
| Expression |
1 | | utopustuq.1 |
. . . . . 6
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) |
2 | | simpl 472 |
. . . . . . . . . . 11
⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → 𝑝 = 𝑞) |
3 | 2 | sneqd 4137 |
. . . . . . . . . 10
⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → {𝑝} = {𝑞}) |
4 | 3 | imaeq2d 5385 |
. . . . . . . . 9
⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑞})) |
5 | 4 | mpteq2dva 4672 |
. . . . . . . 8
⊢ (𝑝 = 𝑞 → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) |
6 | 5 | rneqd 5274 |
. . . . . . 7
⊢ (𝑝 = 𝑞 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) |
7 | 6 | cbvmptv 4678 |
. . . . . 6
⊢ (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) = (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) |
8 | 1, 7 | eqtri 2632 |
. . . . 5
⊢ 𝑁 = (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) |
9 | 8 | a1i 11 |
. . . 4
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → 𝑁 = (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))) |
10 | | simpr2 1061 |
. . . . . . . . 9
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → 𝑞 = 𝑃) |
11 | 10 | sneqd 4137 |
. . . . . . . 8
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → {𝑞} = {𝑃}) |
12 | 11 | imaeq2d 5385 |
. . . . . . 7
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃})) |
13 | 12 | 3anassrs 1282 |
. . . . . 6
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃})) |
14 | 13 | mpteq2dva 4672 |
. . . . 5
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
15 | 14 | rneqd 5274 |
. . . 4
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
16 | | simpr 476 |
. . . 4
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋) |
17 | | mptexg 6389 |
. . . . . 6
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) |
18 | | rnexg 6990 |
. . . . . 6
⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) |
19 | 17, 18 | syl 17 |
. . . . 5
⊢ (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) |
20 | 19 | adantr 480 |
. . . 4
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) |
21 | 9, 15, 16, 20 | fvmptd 6197 |
. . 3
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
22 | 21 | eleq2d 2673 |
. 2
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑁‘𝑃) ↔ 𝐴 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))) |
23 | | imaeq1 5380 |
. . . 4
⊢ (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃})) |
24 | 23 | cbvmptv 4678 |
. . 3
⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑤 ∈ 𝑈 ↦ (𝑤 “ {𝑃})) |
25 | 24 | elrnmpt 5293 |
. 2
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃}))) |
26 | 22, 25 | sylan9bb 732 |
1
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑁‘𝑃) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃}))) |