Proof of Theorem gruima
Step | Hyp | Ref
| Expression |
1 | | simpl2 1192 |
. . . 4
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → Fun 𝐹) |
2 | | funrel 6554 |
. . . 4
⊢ (Fun
𝐹 → Rel 𝐹) |
3 | | df-ima 5682 |
. . . . 5
⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) |
4 | | resres 5986 |
. . . . . . 7
⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) |
5 | | resdm 6018 |
. . . . . . . 8
⊢ (Rel
𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) |
6 | 5 | reseq1d 5972 |
. . . . . . 7
⊢ (Rel
𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴)) |
7 | 4, 6 | eqtr3id 2785 |
. . . . . 6
⊢ (Rel
𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴)) |
8 | 7 | rneqd 5929 |
. . . . 5
⊢ (Rel
𝐹 → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = ran (𝐹 ↾ 𝐴)) |
9 | 3, 8 | eqtr4id 2790 |
. . . 4
⊢ (Rel
𝐹 → (𝐹 “ 𝐴) = ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴))) |
10 | 1, 2, 9 | 3syl 18 |
. . 3
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) = ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴))) |
11 | | simpl1 1191 |
. . . 4
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝑈 ∈ Univ) |
12 | | simpr 485 |
. . . . 5
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝑈) |
13 | | inss2 4225 |
. . . . . 6
⊢ (dom
𝐹 ∩ 𝐴) ⊆ 𝐴 |
14 | 13 | a1i 11 |
. . . . 5
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (dom 𝐹 ∩ 𝐴) ⊆ 𝐴) |
15 | | gruss 10773 |
. . . . 5
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (dom 𝐹 ∩ 𝐴) ⊆ 𝐴) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈) |
16 | 11, 12, 14, 15 | syl3anc 1371 |
. . . 4
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈) |
17 | | funforn 6799 |
. . . . . . . 8
⊢ (Fun
𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹) |
18 | | fof 6792 |
. . . . . . . 8
⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) |
19 | 17, 18 | sylbi 216 |
. . . . . . 7
⊢ (Fun
𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) |
20 | | inss1 4224 |
. . . . . . 7
⊢ (dom
𝐹 ∩ 𝐴) ⊆ dom 𝐹 |
21 | | fssres 6744 |
. . . . . . 7
⊢ ((𝐹:dom 𝐹⟶ran 𝐹 ∧ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹) |
22 | 19, 20, 21 | sylancl 586 |
. . . . . 6
⊢ (Fun
𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹) |
23 | | ffn 6704 |
. . . . . 6
⊢ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴)) |
24 | 1, 22, 23 | 3syl 18 |
. . . . 5
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴)) |
25 | | simpl3 1193 |
. . . . . 6
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ⊆ 𝑈) |
26 | 10, 25 | eqsstrrd 4017 |
. . . . 5
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ⊆ 𝑈) |
27 | | df-f 6536 |
. . . . 5
⊢ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈 ↔ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴) ∧ ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ⊆ 𝑈)) |
28 | 24, 26, 27 | sylanbrc 583 |
. . . 4
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈) |
29 | | grurn 10778 |
. . . 4
⊢ ((𝑈 ∈ Univ ∧ (dom 𝐹 ∩ 𝐴) ∈ 𝑈 ∧ (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ∈ 𝑈) |
30 | 11, 16, 28, 29 | syl3anc 1371 |
. . 3
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ∈ 𝑈) |
31 | 10, 30 | eqeltrd 2832 |
. 2
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ∈ 𝑈) |
32 | 31 | ex 413 |
1
⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈)) |