Proof of Theorem gruima
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl2 1193 | . . . 4
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → Fun 𝐹) | 
| 2 |  | funrel 6583 | . . . 4
⊢ (Fun
𝐹 → Rel 𝐹) | 
| 3 |  | df-ima 5698 | . . . . 5
⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | 
| 4 |  | resres 6010 | . . . . . . 7
⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) | 
| 5 |  | resdm 6044 | . . . . . . . 8
⊢ (Rel
𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | 
| 6 | 5 | reseq1d 5996 | . . . . . . 7
⊢ (Rel
𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴)) | 
| 7 | 4, 6 | eqtr3id 2791 | . . . . . 6
⊢ (Rel
𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴)) | 
| 8 | 7 | rneqd 5949 | . . . . 5
⊢ (Rel
𝐹 → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = ran (𝐹 ↾ 𝐴)) | 
| 9 | 3, 8 | eqtr4id 2796 | . . . 4
⊢ (Rel
𝐹 → (𝐹 “ 𝐴) = ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴))) | 
| 10 | 1, 2, 9 | 3syl 18 | . . 3
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) = ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴))) | 
| 11 |  | simpl1 1192 | . . . 4
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝑈 ∈ Univ) | 
| 12 |  | simpr 484 | . . . . 5
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝑈) | 
| 13 |  | inss2 4238 | . . . . . 6
⊢ (dom
𝐹 ∩ 𝐴) ⊆ 𝐴 | 
| 14 | 13 | a1i 11 | . . . . 5
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (dom 𝐹 ∩ 𝐴) ⊆ 𝐴) | 
| 15 |  | gruss 10836 | . . . . 5
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (dom 𝐹 ∩ 𝐴) ⊆ 𝐴) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈) | 
| 16 | 11, 12, 14, 15 | syl3anc 1373 | . . . 4
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈) | 
| 17 |  | funforn 6827 | . . . . . . . 8
⊢ (Fun
𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹) | 
| 18 |  | fof 6820 | . . . . . . . 8
⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) | 
| 19 | 17, 18 | sylbi 217 | . . . . . . 7
⊢ (Fun
𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) | 
| 20 |  | inss1 4237 | . . . . . . 7
⊢ (dom
𝐹 ∩ 𝐴) ⊆ dom 𝐹 | 
| 21 |  | fssres 6774 | . . . . . . 7
⊢ ((𝐹:dom 𝐹⟶ran 𝐹 ∧ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹) | 
| 22 | 19, 20, 21 | sylancl 586 | . . . . . 6
⊢ (Fun
𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹) | 
| 23 |  | ffn 6736 | . . . . . 6
⊢ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴)) | 
| 24 | 1, 22, 23 | 3syl 18 | . . . . 5
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴)) | 
| 25 |  | simpl3 1194 | . . . . . 6
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ⊆ 𝑈) | 
| 26 | 10, 25 | eqsstrrd 4019 | . . . . 5
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ⊆ 𝑈) | 
| 27 |  | df-f 6565 | . . . . 5
⊢ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈 ↔ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴) ∧ ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ⊆ 𝑈)) | 
| 28 | 24, 26, 27 | sylanbrc 583 | . . . 4
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈) | 
| 29 |  | grurn 10841 | . . . 4
⊢ ((𝑈 ∈ Univ ∧ (dom 𝐹 ∩ 𝐴) ∈ 𝑈 ∧ (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ∈ 𝑈) | 
| 30 | 11, 16, 28, 29 | syl3anc 1373 | . . 3
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ∈ 𝑈) | 
| 31 | 10, 30 | eqeltrd 2841 | . 2
⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ∈ 𝑈) | 
| 32 | 31 | ex 412 | 1
⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈)) |