| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | marypha2lem.t | . . . . . 6
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) | 
| 2 | 1 | marypha2lem2 9476 | . . . . 5
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} | 
| 3 | 2 | imaeq1i 6075 | . . . 4
⊢ (𝑇 “ 𝑋) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋) | 
| 4 |  | df-ima 5698 | . . . 4
⊢
({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} “ 𝑋) = ran ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) | 
| 5 | 3, 4 | eqtri 2765 | . . 3
⊢ (𝑇 “ 𝑋) = ran ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) | 
| 6 |  | resopab2 6054 | . . . . . 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 5949 | . . . 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 5965 | . . . . 5
⊢ ran
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} | 
| 10 |  | df-rex 3071 | . . . . . . . . 9
⊢
(∃𝑥 ∈
𝑋 𝑦 ∈ (𝐹‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))) | 
| 11 | 10 | bicomi 224 | . . . . . . . 8
⊢
(∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥)) ↔ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)) | 
| 12 | 11 | abbii 2809 | . . . . . . 7
⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)} | 
| 13 |  | df-iun 4993 | . . . . . . 7
⊢ ∪ 𝑥 ∈ 𝑋 (𝐹‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 ∈ (𝐹‘𝑥)} | 
| 14 | 12, 13 | eqtr4i 2768 | . . . . . 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 | eqtrid 2789 | . . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝐹‘𝑥))} = ∪
𝑥 ∈ 𝑋 (𝐹‘𝑥)) | 
| 17 | 8, 16 | eqtrd 2777 | . . 3
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ran ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} ↾ 𝑋) = ∪
𝑥 ∈ 𝑋 (𝐹‘𝑥)) | 
| 18 | 5, 17 | eqtrid 2789 | . 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪
𝑥 ∈ 𝑋 (𝐹‘𝑥)) | 
| 19 |  | fnfun 6668 | . . . 4
⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | 
| 20 | 19 | adantr 480 | . . 3
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → Fun 𝐹) | 
| 21 |  | funiunfv 7268 | . . 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 2777 | 1
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋)) |