| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sneq 4635 | . . . . . 6
⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵}) | 
| 2 |  | reseq2 5991 | . . . . . . . 8
⊢ ({𝑥} = {𝐵} → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵})) | 
| 3 | 2 | feq1d 6719 | . . . . . . 7
⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝑥}⟶𝐶)) | 
| 4 |  | feq2 6716 | . . . . . . 7
⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝐵}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶)) | 
| 5 | 3, 4 | bitrd 279 | . . . . . 6
⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶)) | 
| 6 | 1, 5 | syl 17 | . . . . 5
⊢ (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶)) | 
| 7 |  | fveq2 6905 | . . . . . 6
⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) | 
| 8 | 7 | eleq1d 2825 | . . . . 5
⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)) | 
| 9 | 6, 8 | bibi12d 345 | . . . 4
⊢ (𝑥 = 𝐵 → (((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶) ↔ ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))) | 
| 10 | 9 | imbi2d 340 | . . 3
⊢ (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) ↔ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)))) | 
| 11 |  | fnressn 7177 | . . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {〈𝑥, (𝐹‘𝑥)〉}) | 
| 12 |  | vsnid 4662 | . . . . . . . . . 10
⊢ 𝑥 ∈ {𝑥} | 
| 13 |  | fvres 6924 | . . . . . . . . . 10
⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)) | 
| 14 | 12, 13 | ax-mp 5 | . . . . . . . . 9
⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥) | 
| 15 | 14 | opeq2i 4876 | . . . . . . . 8
⊢
〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉 = 〈𝑥, (𝐹‘𝑥)〉 | 
| 16 | 15 | sneqi 4636 | . . . . . . 7
⊢
{〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} = {〈𝑥, (𝐹‘𝑥)〉} | 
| 17 | 16 | eqeq2i 2749 | . . . . . 6
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} ↔ (𝐹 ↾ {𝑥}) = {〈𝑥, (𝐹‘𝑥)〉}) | 
| 18 |  | vex 3483 | . . . . . . . 8
⊢ 𝑥 ∈ V | 
| 19 | 18 | fsn2 7155 | . . . . . . 7
⊢ ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉})) | 
| 20 |  | iba 527 | . . . . . . . 8
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} → (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉}))) | 
| 21 | 14 | eleq1i 2831 | . . . . . . . 8
⊢ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶) | 
| 22 | 20, 21 | bitr3di 286 | . . . . . . 7
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} → ((((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉}) ↔ (𝐹‘𝑥) ∈ 𝐶)) | 
| 23 | 19, 22 | bitrid 283 | . . . . . 6
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) | 
| 24 | 17, 23 | sylbir 235 | . . . . 5
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, (𝐹‘𝑥)〉} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) | 
| 25 | 11, 24 | syl 17 | . . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) | 
| 26 | 25 | expcom 413 | . . 3
⊢ (𝑥 ∈ 𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))) | 
| 27 | 10, 26 | vtoclga 3576 | . 2
⊢ (𝐵 ∈ 𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))) | 
| 28 | 27 | impcom 407 | 1
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)) |