Step | Hyp | Ref
| Expression |
1 | | marypha2lem.t |
. . . . . 6
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) |
2 | 1 | marypha2lem2 8225 |
. . . . 5
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} |
3 | 2 | imaeq1i 5382 |
. . . 4
⊢ (𝑇 “ 𝑋) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋) |
4 | | df-ima 5051 |
. . . 4
⊢
({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋) = ran ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) |
5 | 3, 4 | eqtri 2632 |
. . 3
⊢ (𝑇 “ 𝑋) = ran ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) |
6 | | resopab2 5368 |
. . . . . 6
⊢ (𝑋 ⊆ 𝐴 → ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))}) |
7 | 6 | adantl 481 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))}) |
8 | 7 | rneqd 5274 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))}) |
9 | | rnopab 5291 |
. . . . 5
⊢ ran
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} |
10 | | df-rex 2902 |
. . . . . . . . 9
⊢
(∃𝑥 ∈
𝑋 𝑦 ∈ (𝐹‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))) |
11 | 10 | bicomi 213 |
. . . . . . . 8
⊢
(∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)) |
12 | 11 | abbii 2726 |
. . . . . . 7
⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)} |
13 | | df-iun 4457 |
. . . . . . 7
⊢ ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)} |
14 | 12, 13 | eqtr4i 2635 |
. . . . . 6
⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪
𝑥 ∈ 𝑋 (𝐹‘𝑥) |
15 | 14 | a1i 11 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪
𝑥 ∈ 𝑋 (𝐹‘𝑥)) |
16 | 9, 15 | syl5eq 2656 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪
𝑥 ∈ 𝑋 (𝐹‘𝑥)) |
17 | 8, 16 | eqtrd 2644 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = ∪
𝑥 ∈ 𝑋 (𝐹‘𝑥)) |
18 | 5, 17 | syl5eq 2656 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪
𝑥 ∈ 𝑋 (𝐹‘𝑥)) |
19 | | fnfun 5902 |
. . . 4
⊢ (𝐹 Fn 𝐴 → Fun 𝐹) |
20 | 19 | adantr 480 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → Fun 𝐹) |
21 | | funiunfv 6410 |
. . 3
⊢ (Fun
𝐹 → ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = ∪ (𝐹 “ 𝑋)) |
22 | 20, 21 | syl 17 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ∪
𝑥 ∈ 𝑋 (𝐹‘𝑥) = ∪ (𝐹 “ 𝑋)) |
23 | 18, 22 | eqtrd 2644 |
1
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋)) |