Proof of Theorem f1dmvrnfibi
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | rnfi 9381 | . 2
⊢ (𝐹 ∈ Fin → ran 𝐹 ∈ Fin) | 
| 2 |  | simpr 484 | . . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → ran 𝐹 ∈ Fin) | 
| 3 |  | f1dm 6807 | . . . . . . . . 9
⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴) | 
| 4 |  | f1f1orn 6858 | . . . . . . . . 9
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran
𝐹) | 
| 5 |  | eleq1 2828 | . . . . . . . . . . . . 13
⊢ (𝐴 = dom 𝐹 → (𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉)) | 
| 6 |  | f1oeq2 6836 | . . . . . . . . . . . . 13
⊢ (𝐴 = dom 𝐹 → (𝐹:𝐴–1-1-onto→ran
𝐹 ↔ 𝐹:dom 𝐹–1-1-onto→ran
𝐹)) | 
| 7 | 5, 6 | anbi12d 632 | . . . . . . . . . . . 12
⊢ (𝐴 = dom 𝐹 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→ran
𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran
𝐹))) | 
| 8 | 7 | eqcoms 2744 | . . . . . . . . . . 11
⊢ (dom
𝐹 = 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→ran
𝐹) ↔ (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran
𝐹))) | 
| 9 | 8 | biimpd 229 | . . . . . . . . . 10
⊢ (dom
𝐹 = 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→ran
𝐹) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran
𝐹))) | 
| 10 | 9 | expcomd 416 | . . . . . . . . 9
⊢ (dom
𝐹 = 𝐴 → (𝐹:𝐴–1-1-onto→ran
𝐹 → (𝐴 ∈ 𝑉 → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran
𝐹)))) | 
| 11 | 3, 4, 10 | sylc 65 | . . . . . . . 8
⊢ (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ 𝑉 → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran
𝐹))) | 
| 12 | 11 | impcom 407 | . . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran
𝐹)) | 
| 13 | 12 | adantr 480 | . . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → (dom 𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran
𝐹)) | 
| 14 |  | f1oeng 9012 | . . . . . 6
⊢ ((dom
𝐹 ∈ 𝑉 ∧ 𝐹:dom 𝐹–1-1-onto→ran
𝐹) → dom 𝐹 ≈ ran 𝐹) | 
| 15 | 13, 14 | syl 17 | . . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ≈ ran 𝐹) | 
| 16 |  | enfii 9227 | . . . . 5
⊢ ((ran
𝐹 ∈ Fin ∧ dom
𝐹 ≈ ran 𝐹) → dom 𝐹 ∈ Fin) | 
| 17 | 2, 15, 16 | syl2anc 584 | . . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → dom 𝐹 ∈ Fin) | 
| 18 |  | f1fun 6805 | . . . . . 6
⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹) | 
| 19 | 18 | ad2antlr 727 | . . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → Fun 𝐹) | 
| 20 |  | fundmfibi 9377 | . . . . 5
⊢ (Fun
𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin)) | 
| 21 | 19, 20 | syl 17 | . . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin)) | 
| 22 | 17, 21 | mpbird 257 | . . 3
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ ran 𝐹 ∈ Fin) → 𝐹 ∈ Fin) | 
| 23 | 22 | ex 412 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ∈ Fin → 𝐹 ∈ Fin)) | 
| 24 | 1, 23 | impbid2 226 | 1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin)) |