Step | Hyp | Ref
| Expression |
1 | | dfss2 2934 |
. 2
⊢ ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵)) |
2 | | eqcom 2042 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) |
3 | | ssel 2939 |
. . . . . . . . . . . 12
⊢ (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹)) |
4 | | funbrfvb 5216 |
. . . . . . . . . . . . 13
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) |
5 | 4 | ex 108 |
. . . . . . . . . . . 12
⊢ (Fun
𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))) |
6 | 3, 5 | syl9 66 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ dom 𝐹 → (Fun 𝐹 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)))) |
7 | 6 | imp31 243 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) |
8 | 2, 7 | syl5bb 181 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)) |
9 | 8 | rexbidva 2323 |
. . . . . . . 8
⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦)) |
10 | | vex 2560 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
11 | 10 | elima 4673 |
. . . . . . . 8
⊢ (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦) |
12 | 9, 11 | syl6rbbr 188 |
. . . . . . 7
⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) |
13 | 12 | imbi1d 220 |
. . . . . 6
⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → ((𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))) |
14 | | r19.23v 2425 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)) |
15 | 13, 14 | syl6bbr 187 |
. . . . 5
⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → ((𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))) |
16 | 15 | albidv 1705 |
. . . 4
⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))) |
17 | 16 | ancoms 255 |
. . 3
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))) |
18 | | ralcom4 2576 |
. . . 4
⊢
(∀𝑥 ∈
𝐴 ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)) |
19 | | ssel2 2940 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹) |
20 | 19 | anim2i 324 |
. . . . . . . 8
⊢ ((Fun
𝐹 ∧ (𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴)) → (Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹)) |
21 | 20 | 3impb 1100 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹)) |
22 | | funfvex 5192 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V) |
23 | | nfv 1421 |
. . . . . . . 8
⊢
Ⅎ𝑦(𝐹‘𝑥) ∈ 𝐵 |
24 | | eleq1 2100 |
. . . . . . . 8
⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) |
25 | 23, 24 | ceqsalg 2582 |
. . . . . . 7
⊢ ((𝐹‘𝑥) ∈ V → (∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ (𝐹‘𝑥) ∈ 𝐵)) |
26 | 21, 22, 25 | 3syl 17 |
. . . . . 6
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ (𝐹‘𝑥) ∈ 𝐵)) |
27 | 26 | 3expa 1104 |
. . . . 5
⊢ (((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑥 ∈ 𝐴) → (∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ (𝐹‘𝑥) ∈ 𝐵)) |
28 | 27 | ralbidva 2322 |
. . . 4
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
29 | 18, 28 | syl5bbr 183 |
. . 3
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
30 | 17, 29 | bitrd 177 |
. 2
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
31 | 1, 30 | syl5bb 181 |
1
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |