Proof of Theorem fompt
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fof 6820 | . . . 4
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | 
| 2 |  | fompt.1 | . . . . 5
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | 
| 3 | 2 | fmpt 7130 | . . . 4
⊢
(∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) | 
| 4 | 1, 3 | sylibr 234 | . . 3
⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) | 
| 5 |  | nfmpt1 5250 | . . . . . . 7
⊢
Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶) | 
| 6 | 2, 5 | nfcxfr 2903 | . . . . . 6
⊢
Ⅎ𝑥𝐹 | 
| 7 | 6 | foelrnf 7128 | . . . . 5
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) | 
| 8 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑥𝐴 | 
| 9 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑥𝐵 | 
| 10 | 6, 8, 9 | nffo 6819 | . . . . . . 7
⊢
Ⅎ𝑥 𝐹:𝐴–onto→𝐵 | 
| 11 |  | simpr 484 | . . . . . . . . 9
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥)) | 
| 12 |  | simpr 484 | . . . . . . . . . . 11
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | 
| 13 | 4 | r19.21bi 3251 | . . . . . . . . . . 11
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | 
| 14 | 2 | fvmpt2 7027 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝑥) = 𝐶) | 
| 15 | 12, 13, 14 | syl2anc 584 | . . . . . . . . . 10
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) | 
| 16 | 15 | adantr 480 | . . . . . . . . 9
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → (𝐹‘𝑥) = 𝐶) | 
| 17 | 11, 16 | eqtrd 2777 | . . . . . . . 8
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = 𝐶) | 
| 18 | 17 | exp31 419 | . . . . . . 7
⊢ (𝐹:𝐴–onto→𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = (𝐹‘𝑥) → 𝑦 = 𝐶))) | 
| 19 | 10, 18 | reximdai 3261 | . . . . . 6
⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)) | 
| 20 | 19 | adantr 480 | . . . . 5
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)) | 
| 21 | 7, 20 | mpd 15 | . . . 4
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) | 
| 22 | 21 | ralrimiva 3146 | . . 3
⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) | 
| 23 | 4, 22 | jca 511 | . 2
⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)) | 
| 24 | 3 | biimpi 216 | . . . 4
⊢
(∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 → 𝐹:𝐴⟶𝐵) | 
| 25 | 24 | adantr 480 | . . 3
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴⟶𝐵) | 
| 26 |  | nfv 1914 | . . . . 5
⊢
Ⅎ𝑦∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 | 
| 27 |  | nfra1 3284 | . . . . 5
⊢
Ⅎ𝑦∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶 | 
| 28 | 26, 27 | nfan 1899 | . . . 4
⊢
Ⅎ𝑦(∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) | 
| 29 |  | simpll 767 | . . . . 5
⊢
(((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) | 
| 30 |  | rspa 3248 | . . . . . 6
⊢
((∀𝑦 ∈
𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) | 
| 31 | 30 | adantll 714 | . . . . 5
⊢
(((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) | 
| 32 |  | nfra1 3284 | . . . . . 6
⊢
Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 | 
| 33 |  | simp3 1139 | . . . . . . . 8
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = 𝐶) | 
| 34 |  | simpr 484 | . . . . . . . . . . 11
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | 
| 35 |  | rspa 3248 | . . . . . . . . . . 11
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | 
| 36 | 34, 35, 14 | syl2anc 584 | . . . . . . . . . 10
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) | 
| 37 | 36 | eqcomd 2743 | . . . . . . . . 9
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 = (𝐹‘𝑥)) | 
| 38 | 37 | 3adant3 1133 | . . . . . . . 8
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐶 = (𝐹‘𝑥)) | 
| 39 | 33, 38 | eqtrd 2777 | . . . . . . 7
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = (𝐹‘𝑥)) | 
| 40 | 39 | 3exp 1120 | . . . . . 6
⊢
(∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → 𝑦 = (𝐹‘𝑥)))) | 
| 41 | 32, 40 | reximdai 3261 | . . . . 5
⊢
(∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) | 
| 42 | 29, 31, 41 | sylc 65 | . . . 4
⊢
(((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) | 
| 43 | 28, 42 | ralrimia 3258 | . . 3
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) | 
| 44 | 6 | dffo3f 7126 | . . 3
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) | 
| 45 | 25, 43, 44 | sylanbrc 583 | . 2
⊢
((∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴–onto→𝐵) | 
| 46 | 23, 45 | impbii 209 | 1
⊢ (𝐹:𝐴–onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)) |