Proof of Theorem ffvresb
Step | Hyp | Ref
| Expression |
1 | | fdm 5325 |
. . . . . 6
⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → dom (𝐹 ↾ 𝐴) = 𝐴) |
2 | | dmres 4887 |
. . . . . . 7
⊢ dom
(𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹) |
3 | | inss2 3328 |
. . . . . . 7
⊢ (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹 |
4 | 2, 3 | eqsstri 3160 |
. . . . . 6
⊢ dom
(𝐹 ↾ 𝐴) ⊆ dom 𝐹 |
5 | 1, 4 | eqsstrrdi 3181 |
. . . . 5
⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹) |
6 | 5 | sselda 3128 |
. . . 4
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹) |
7 | | fvres 5492 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
8 | 7 | adantl 275 |
. . . . 5
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
9 | | ffvelrn 5600 |
. . . . 5
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵) |
10 | 8, 9 | eqeltrrd 2235 |
. . . 4
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
11 | 6, 10 | jca 304 |
. . 3
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) |
12 | 11 | ralrimiva 2530 |
. 2
⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) |
13 | | simpl 108 |
. . . . . . 7
⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝑥 ∈ dom 𝐹) |
14 | 13 | ralimi 2520 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝐹) |
15 | | dfss3 3118 |
. . . . . 6
⊢ (𝐴 ⊆ dom 𝐹 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝐹) |
16 | 14, 15 | sylibr 133 |
. . . . 5
⊢
(∀𝑥 ∈
𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝐴 ⊆ dom 𝐹) |
17 | | funfn 5200 |
. . . . . 6
⊢ (Fun
𝐹 ↔ 𝐹 Fn dom 𝐹) |
18 | | fnssres 5283 |
. . . . . 6
⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴) Fn 𝐴) |
19 | 17, 18 | sylanb 282 |
. . . . 5
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴) Fn 𝐴) |
20 | 16, 19 | sylan2 284 |
. . . 4
⊢ ((Fun
𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → (𝐹 ↾ 𝐴) Fn 𝐴) |
21 | | simpr 109 |
. . . . . . . 8
⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ 𝐵) |
22 | 7 | eleq1d 2226 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → (((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) |
23 | 21, 22 | syl5ibr 155 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵)) |
24 | 23 | ralimia 2518 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵) |
25 | 24 | adantl 275 |
. . . . 5
⊢ ((Fun
𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵) |
26 | | fnfvrnss 5627 |
. . . . 5
⊢ (((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵) → ran (𝐹 ↾ 𝐴) ⊆ 𝐵) |
27 | 20, 25, 26 | syl2anc 409 |
. . . 4
⊢ ((Fun
𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → ran (𝐹 ↾ 𝐴) ⊆ 𝐵) |
28 | | df-f 5174 |
. . . 4
⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) ⊆ 𝐵)) |
29 | 20, 27, 28 | sylanbrc 414 |
. . 3
⊢ ((Fun
𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵) |
30 | 29 | ex 114 |
. 2
⊢ (Fun
𝐹 → (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝐹 ↾ 𝐴):𝐴⟶𝐵)) |
31 | 12, 30 | impbid2 142 |
1
⊢ (Fun
𝐹 → ((𝐹 ↾ 𝐴):𝐴⟶𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))) |