| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | relres 6022 | . 2
⊢ Rel
(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V
×  Singletons ))) | 
| 2 |  | vex 3483 | . . . . . . 7
⊢ 𝑧 ∈ V | 
| 3 | 2 | brresi 6005 | . . . . . 6
⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑧 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )) ∧ 𝑥𝐹𝑧)) | 
| 4 | 3 | simprbi 496 | . . . . 5
⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑧 → 𝑥𝐹𝑧) | 
| 5 |  | vex 3483 | . . . . . . . 8
⊢ 𝑦 ∈ V | 
| 6 | 5 | brresi 6005 | . . . . . . 7
⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑦 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )) ∧ 𝑥𝐹𝑦)) | 
| 7 |  | funpartlem 35944 | . . . . . . . 8
⊢ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V
×  Singletons )) ↔ ∃𝑤(𝐹 “ {𝑥}) = {𝑤}) | 
| 8 | 7 | anbi1i 624 | . . . . . . 7
⊢ ((𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V
×  Singletons )) ∧ 𝑥𝐹𝑦) ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦)) | 
| 9 | 6, 8 | bitri 275 | . . . . . 6
⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑦 ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦)) | 
| 10 |  | df-br 5143 | . . . . . . . . . . 11
⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | 
| 11 |  | df-br 5143 | . . . . . . . . . . 11
⊢ (𝑥𝐹𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝐹) | 
| 12 | 10, 11 | anbi12i 628 | . . . . . . . . . 10
⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹)) | 
| 13 |  | vex 3483 | . . . . . . . . . . . 12
⊢ 𝑥 ∈ V | 
| 14 | 13, 5 | elimasn 6107 | . . . . . . . . . . 11
⊢ (𝑦 ∈ (𝐹 “ {𝑥}) ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | 
| 15 | 13, 2 | elimasn 6107 | . . . . . . . . . . 11
⊢ (𝑧 ∈ (𝐹 “ {𝑥}) ↔ 〈𝑥, 𝑧〉 ∈ 𝐹) | 
| 16 | 14, 15 | anbi12i 628 | . . . . . . . . . 10
⊢ ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹)) | 
| 17 | 12, 16 | bitr4i 278 | . . . . . . . . 9
⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥}))) | 
| 18 |  | eleq2 2829 | . . . . . . . . . . 11
⊢ ((𝐹 “ {𝑥}) = {𝑤} → (𝑦 ∈ (𝐹 “ {𝑥}) ↔ 𝑦 ∈ {𝑤})) | 
| 19 |  | eleq2 2829 | . . . . . . . . . . 11
⊢ ((𝐹 “ {𝑥}) = {𝑤} → (𝑧 ∈ (𝐹 “ {𝑥}) ↔ 𝑧 ∈ {𝑤})) | 
| 20 | 18, 19 | anbi12d 632 | . . . . . . . . . 10
⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤}))) | 
| 21 |  | velsn 4641 | . . . . . . . . . . 11
⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤) | 
| 22 |  | velsn 4641 | . . . . . . . . . . 11
⊢ (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤) | 
| 23 |  | equtr2 2025 | . . . . . . . . . . 11
⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑤) → 𝑦 = 𝑧) | 
| 24 | 21, 22, 23 | syl2anb 598 | . . . . . . . . . 10
⊢ ((𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤}) → 𝑦 = 𝑧) | 
| 25 | 20, 24 | biimtrdi 253 | . . . . . . . . 9
⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) → 𝑦 = 𝑧)) | 
| 26 | 17, 25 | biimtrid 242 | . . . . . . . 8
⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)) | 
| 27 | 26 | exlimiv 1929 | . . . . . . 7
⊢
(∃𝑤(𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)) | 
| 28 | 27 | impl 455 | . . . . . 6
⊢
(((∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦) ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) | 
| 29 | 9, 28 | sylanb 581 | . . . . 5
⊢ ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) | 
| 30 | 4, 29 | sylan2 593 | . . . 4
⊢ ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑧) → 𝑦 = 𝑧) | 
| 31 | 30 | gen2 1795 | . . 3
⊢
∀𝑦∀𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑧) → 𝑦 = 𝑧) | 
| 32 | 31 | ax-gen 1794 | . 2
⊢
∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑧) → 𝑦 = 𝑧) | 
| 33 |  | df-funpart 35876 | . . . 4
⊢
Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V
×  Singletons ))) | 
| 34 | 33 | funeqi 6586 | . . 3
⊢ (Fun
Funpart𝐹 ↔ Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V
×  Singletons )))) | 
| 35 |  | dffun2 6570 | . . 3
⊢ (Fun
(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V
×  Singletons ))) ↔ (Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V
×  Singletons ))) ∧ ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑧) → 𝑦 = 𝑧))) | 
| 36 | 34, 35 | bitri 275 | . 2
⊢ (Fun
Funpart𝐹 ↔ (Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V
×  Singletons ))) ∧ ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V ×  Singletons )))𝑧) → 𝑦 = 𝑧))) | 
| 37 | 1, 32, 36 | mpbir2an 711 | 1
⊢ Fun
Funpart𝐹 |