| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ovexd 7467 | . 2
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m (𝐴 ∪ 𝐵)) ∈ V) | 
| 2 |  | ovexd 7467 | . . 3
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m 𝐴) ∈ V) | 
| 3 |  | ovexd 7467 | . . 3
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m 𝐵) ∈ V) | 
| 4 | 2, 3 | xpexd 7772 | . 2
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) ∈ V) | 
| 5 |  | elmapi 8890 | . . . . 5
⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → 𝑥:(𝐴 ∪ 𝐵)⟶𝐶) | 
| 6 |  | ssun1 4177 | . . . . 5
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | 
| 7 |  | fssres 6773 | . . . . 5
⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶) | 
| 8 | 5, 6, 7 | sylancl 586 | . . . 4
⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶) | 
| 9 |  | ssun2 4178 | . . . . 5
⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | 
| 10 |  | fssres 6773 | . . . . 5
⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶) | 
| 11 | 5, 9, 10 | sylancl 586 | . . . 4
⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶) | 
| 12 | 8, 11 | jca 511 | . . 3
⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶)) | 
| 13 |  | opelxp 5720 | . . . 4
⊢
(〈(𝑥 ↾
𝐴), (𝑥 ↾ 𝐵)〉 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) ↔ ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑m 𝐴) ∧ (𝑥 ↾ 𝐵) ∈ (𝐶 ↑m 𝐵))) | 
| 14 |  | simpl3 1193 | . . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐶 ∈ 𝑋) | 
| 15 |  | simpl1 1191 | . . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ 𝑉) | 
| 16 | 14, 15 | elmapd 8881 | . . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑m 𝐴) ↔ (𝑥 ↾ 𝐴):𝐴⟶𝐶)) | 
| 17 |  | simpl2 1192 | . . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ 𝑊) | 
| 18 | 14, 17 | elmapd 8881 | . . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐵) ∈ (𝐶 ↑m 𝐵) ↔ (𝑥 ↾ 𝐵):𝐵⟶𝐶)) | 
| 19 | 16, 18 | anbi12d 632 | . . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (((𝑥 ↾ 𝐴) ∈ (𝐶 ↑m 𝐴) ∧ (𝑥 ↾ 𝐵) ∈ (𝐶 ↑m 𝐵)) ↔ ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶))) | 
| 20 | 13, 19 | bitrid 283 | . . 3
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) ↔ ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶))) | 
| 21 | 12, 20 | imbitrrid 246 | . 2
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) | 
| 22 |  | xp1st 8047 | . . . . . . 7
⊢ (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → (1st ‘𝑦) ∈ (𝐶 ↑m 𝐴)) | 
| 23 | 22 | adantl 481 | . . . . . 6
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (1st ‘𝑦) ∈ (𝐶 ↑m 𝐴)) | 
| 24 |  | elmapi 8890 | . . . . . 6
⊢
((1st ‘𝑦) ∈ (𝐶 ↑m 𝐴) → (1st ‘𝑦):𝐴⟶𝐶) | 
| 25 | 23, 24 | syl 17 | . . . . 5
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (1st ‘𝑦):𝐴⟶𝐶) | 
| 26 |  | xp2nd 8048 | . . . . . . 7
⊢ (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → (2nd ‘𝑦) ∈ (𝐶 ↑m 𝐵)) | 
| 27 | 26 | adantl 481 | . . . . . 6
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (2nd ‘𝑦) ∈ (𝐶 ↑m 𝐵)) | 
| 28 |  | elmapi 8890 | . . . . . 6
⊢
((2nd ‘𝑦) ∈ (𝐶 ↑m 𝐵) → (2nd ‘𝑦):𝐵⟶𝐶) | 
| 29 | 27, 28 | syl 17 | . . . . 5
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (2nd ‘𝑦):𝐵⟶𝐶) | 
| 30 |  | simplr 768 | . . . . 5
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (𝐴 ∩ 𝐵) = ∅) | 
| 31 | 25, 29, 30 | fun2d 6771 | . . . 4
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → ((1st ‘𝑦) ∪ (2nd
‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶) | 
| 32 | 31 | ex 412 | . . 3
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → ((1st ‘𝑦) ∪ (2nd
‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶)) | 
| 33 |  | unexg 7764 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | 
| 34 | 15, 17, 33 | syl2anc 584 | . . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ V) | 
| 35 | 14, 34 | elmapd 8881 | . . 3
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (((1st
‘𝑦) ∪
(2nd ‘𝑦))
∈ (𝐶
↑m (𝐴 ∪
𝐵)) ↔ ((1st
‘𝑦) ∪
(2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶)) | 
| 36 | 32, 35 | sylibrd 259 | . 2
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → ((1st ‘𝑦) ∪ (2nd
‘𝑦)) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)))) | 
| 37 |  | 1st2nd2 8054 | . . . . . . 7
⊢ (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | 
| 38 | 37 | ad2antll 729 | . . . . . 6
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | 
| 39 | 25 | adantrl 716 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (1st ‘𝑦):𝐴⟶𝐶) | 
| 40 | 29 | adantrl 716 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (2nd ‘𝑦):𝐵⟶𝐶) | 
| 41 |  | res0 6000 | . . . . . . . . . 10
⊢
((1st ‘𝑦) ↾ ∅) = ∅ | 
| 42 |  | res0 6000 | . . . . . . . . . 10
⊢
((2nd ‘𝑦) ↾ ∅) = ∅ | 
| 43 | 41, 42 | eqtr4i 2767 | . . . . . . . . 9
⊢
((1st ‘𝑦) ↾ ∅) = ((2nd
‘𝑦) ↾
∅) | 
| 44 |  | simplr 768 | . . . . . . . . . 10
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (𝐴 ∩ 𝐵) = ∅) | 
| 45 | 44 | reseq2d 5996 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((1st ‘𝑦) ↾
∅)) | 
| 46 | 44 | reseq2d 5996 | . . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾
∅)) | 
| 47 | 43, 45, 46 | 3eqtr4a 2802 | . . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵))) | 
| 48 |  | fresaunres1 6780 | . . . . . . . 8
⊢
(((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑦) ∪ (2nd
‘𝑦)) ↾ 𝐴) = (1st ‘𝑦)) | 
| 49 | 39, 40, 47, 48 | syl3anc 1372 | . . . . . . 7
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (((1st ‘𝑦) ∪ (2nd
‘𝑦)) ↾ 𝐴) = (1st ‘𝑦)) | 
| 50 |  | fresaunres2 6779 | . . . . . . . 8
⊢
(((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑦) ∪ (2nd
‘𝑦)) ↾ 𝐵) = (2nd ‘𝑦)) | 
| 51 | 39, 40, 47, 50 | syl3anc 1372 | . . . . . . 7
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (((1st ‘𝑦) ∪ (2nd
‘𝑦)) ↾ 𝐵) = (2nd ‘𝑦)) | 
| 52 | 49, 51 | opeq12d 4880 | . . . . . 6
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → 〈(((1st
‘𝑦) ∪
(2nd ‘𝑦))
↾ 𝐴),
(((1st ‘𝑦)
∪ (2nd ‘𝑦)) ↾ 𝐵)〉 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | 
| 53 | 38, 52 | eqtr4d 2779 | . . . . 5
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → 𝑦 = 〈(((1st ‘𝑦) ∪ (2nd
‘𝑦)) ↾ 𝐴), (((1st
‘𝑦) ∪
(2nd ‘𝑦))
↾ 𝐵)〉) | 
| 54 |  | reseq1 5990 | . . . . . . 7
⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd
‘𝑦)) → (𝑥 ↾ 𝐴) = (((1st ‘𝑦) ∪ (2nd
‘𝑦)) ↾ 𝐴)) | 
| 55 |  | reseq1 5990 | . . . . . . 7
⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd
‘𝑦)) → (𝑥 ↾ 𝐵) = (((1st ‘𝑦) ∪ (2nd
‘𝑦)) ↾ 𝐵)) | 
| 56 | 54, 55 | opeq12d 4880 | . . . . . 6
⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd
‘𝑦)) →
〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 = 〈(((1st
‘𝑦) ∪
(2nd ‘𝑦))
↾ 𝐴),
(((1st ‘𝑦)
∪ (2nd ‘𝑦)) ↾ 𝐵)〉) | 
| 57 | 56 | eqeq2d 2747 | . . . . 5
⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd
‘𝑦)) → (𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 ↔ 𝑦 = 〈(((1st ‘𝑦) ∪ (2nd
‘𝑦)) ↾ 𝐴), (((1st
‘𝑦) ∪
(2nd ‘𝑦))
↾ 𝐵)〉)) | 
| 58 | 53, 57 | syl5ibrcom 247 | . . . 4
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → 𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉)) | 
| 59 |  | ffn 6735 | . . . . . . . 8
⊢ (𝑥:(𝐴 ∪ 𝐵)⟶𝐶 → 𝑥 Fn (𝐴 ∪ 𝐵)) | 
| 60 |  | fnresdm 6686 | . . . . . . . 8
⊢ (𝑥 Fn (𝐴 ∪ 𝐵) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥) | 
| 61 | 5, 59, 60 | 3syl 18 | . . . . . . 7
⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥) | 
| 62 | 61 | ad2antrl 728 | . . . . . 6
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥) | 
| 63 | 62 | eqcomd 2742 | . . . . 5
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → 𝑥 = (𝑥 ↾ (𝐴 ∪ 𝐵))) | 
| 64 |  | vex 3483 | . . . . . . . . . 10
⊢ 𝑥 ∈ V | 
| 65 | 64 | resex 6046 | . . . . . . . . 9
⊢ (𝑥 ↾ 𝐴) ∈ V | 
| 66 | 64 | resex 6046 | . . . . . . . . 9
⊢ (𝑥 ↾ 𝐵) ∈ V | 
| 67 | 65, 66 | op1std 8025 | . . . . . . . 8
⊢ (𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 → (1st ‘𝑦) = (𝑥 ↾ 𝐴)) | 
| 68 | 65, 66 | op2ndd 8026 | . . . . . . . 8
⊢ (𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 → (2nd ‘𝑦) = (𝑥 ↾ 𝐵)) | 
| 69 | 67, 68 | uneq12d 4168 | . . . . . . 7
⊢ (𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 → ((1st ‘𝑦) ∪ (2nd
‘𝑦)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵))) | 
| 70 |  | resundi 6010 | . . . . . . 7
⊢ (𝑥 ↾ (𝐴 ∪ 𝐵)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵)) | 
| 71 | 69, 70 | eqtr4di 2794 | . . . . . 6
⊢ (𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 → ((1st ‘𝑦) ∪ (2nd
‘𝑦)) = (𝑥 ↾ (𝐴 ∪ 𝐵))) | 
| 72 | 71 | eqeq2d 2747 | . . . . 5
⊢ (𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴 ∪ 𝐵)))) | 
| 73 | 63, 72 | syl5ibrcom 247 | . . . 4
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉 → 𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)))) | 
| 74 | 58, 73 | impbid 212 | . . 3
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉)) | 
| 75 | 74 | ex 412 | . 2
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑦 = 〈(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)〉))) | 
| 76 | 1, 4, 21, 36, 75 | en3d 9030 | 1
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m (𝐴 ∪ 𝐵)) ≈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) |