Proof of Theorem ffvresb
| Step | Hyp | Ref
| Expression |
| 1 | | fdm 6745 |
. . . . . 6
⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → dom (𝐹 ↾ 𝐴) = 𝐴) |
| 2 | | dmres 6030 |
. . . . . . 7
⊢ dom
(𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹) |
| 3 | | inss2 4238 |
. . . . . . 7
⊢ (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹 |
| 4 | 2, 3 | eqsstri 4030 |
. . . . . 6
⊢ dom
(𝐹 ↾ 𝐴) ⊆ dom 𝐹 |
| 5 | 1, 4 | eqsstrrdi 4029 |
. . . . 5
⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹) |
| 6 | 5 | sselda 3983 |
. . . 4
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹) |
| 7 | | fvres 6925 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
| 8 | 7 | adantl 481 |
. . . . 5
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
| 9 | | ffvelcdm 7101 |
. . . . 5
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵) |
| 10 | 8, 9 | eqeltrrd 2842 |
. . . 4
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
| 11 | 6, 10 | jca 511 |
. . 3
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) |
| 12 | 11 | ralrimiva 3146 |
. 2
⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) |
| 13 | | simpl 482 |
. . . . . . 7
⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝑥 ∈ dom 𝐹) |
| 14 | 13 | ralimi 3083 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝐹) |
| 15 | | dfss3 3972 |
. . . . . 6
⊢ (𝐴 ⊆ dom 𝐹 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝐹) |
| 16 | 14, 15 | sylibr 234 |
. . . . 5
⊢
(∀𝑥 ∈
𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝐴 ⊆ dom 𝐹) |
| 17 | | funfn 6596 |
. . . . . 6
⊢ (Fun
𝐹 ↔ 𝐹 Fn dom 𝐹) |
| 18 | | fnssres 6691 |
. . . . . 6
⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴) Fn 𝐴) |
| 19 | 17, 18 | sylanb 581 |
. . . . 5
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴) Fn 𝐴) |
| 20 | 16, 19 | sylan2 593 |
. . . 4
⊢ ((Fun
𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → (𝐹 ↾ 𝐴) Fn 𝐴) |
| 21 | | simpr 484 |
. . . . . . . 8
⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ 𝐵) |
| 22 | 7 | eleq1d 2826 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → (((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) |
| 23 | 21, 22 | imbitrrid 246 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵)) |
| 24 | 23 | ralimia 3080 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵) |
| 25 | 24 | adantl 481 |
. . . . 5
⊢ ((Fun
𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵) |
| 26 | | fnfvrnss 7141 |
. . . . 5
⊢ (((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵) → ran (𝐹 ↾ 𝐴) ⊆ 𝐵) |
| 27 | 20, 25, 26 | syl2anc 584 |
. . . 4
⊢ ((Fun
𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → ran (𝐹 ↾ 𝐴) ⊆ 𝐵) |
| 28 | | df-f 6565 |
. . . 4
⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) ⊆ 𝐵)) |
| 29 | 20, 27, 28 | sylanbrc 583 |
. . 3
⊢ ((Fun
𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵) |
| 30 | 29 | ex 412 |
. 2
⊢ (Fun
𝐹 → (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝐹 ↾ 𝐴):𝐴⟶𝐵)) |
| 31 | 12, 30 | impbid2 226 |
1
⊢ (Fun
𝐹 → ((𝐹 ↾ 𝐴):𝐴⟶𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))) |