| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fsneq.f | . . 3
⊢ (𝜑 → 𝐹 Fn 𝐵) | 
| 2 |  | fsneq.g | . . 3
⊢ (𝜑 → 𝐺 Fn 𝐵) | 
| 3 |  | eqfnfv 7050 | . . 3
⊢ ((𝐹 Fn 𝐵 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | 
| 4 | 1, 2, 3 | syl2anc 584 | . 2
⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | 
| 5 |  | fsneq.a | . . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| 6 |  | snidg 4659 | . . . . . . . 8
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | 
| 7 | 5, 6 | syl 17 | . . . . . . 7
⊢ (𝜑 → 𝐴 ∈ {𝐴}) | 
| 8 |  | fsneq.b | . . . . . . . . 9
⊢ 𝐵 = {𝐴} | 
| 9 | 8 | eqcomi 2745 | . . . . . . . 8
⊢ {𝐴} = 𝐵 | 
| 10 | 9 | a1i 11 | . . . . . . 7
⊢ (𝜑 → {𝐴} = 𝐵) | 
| 11 | 7, 10 | eleqtrd 2842 | . . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝐵) | 
| 12 | 11 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → 𝐴 ∈ 𝐵) | 
| 13 |  | simpr 484 | . . . . 5
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) | 
| 14 |  | fveq2 6905 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | 
| 15 |  | fveq2 6905 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝐺‘𝑥) = (𝐺‘𝐴)) | 
| 16 | 14, 15 | eqeq12d 2752 | . . . . . 6
⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴))) | 
| 17 | 16 | rspcva 3619 | . . . . 5
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝐹‘𝐴) = (𝐺‘𝐴)) | 
| 18 | 12, 13, 17 | syl2anc 584 | . . . 4
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝐹‘𝐴) = (𝐺‘𝐴)) | 
| 19 | 18 | ex 412 | . . 3
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) → (𝐹‘𝐴) = (𝐺‘𝐴))) | 
| 20 |  | simpl 482 | . . . . . . 7
⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝐴) = (𝐺‘𝐴)) | 
| 21 | 8 | eleq2i 2832 | . . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ {𝐴}) | 
| 22 | 21 | biimpi 216 | . . . . . . . . . 10
⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {𝐴}) | 
| 23 |  | velsn 4641 | . . . . . . . . . 10
⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | 
| 24 | 22, 23 | sylib 218 | . . . . . . . . 9
⊢ (𝑥 ∈ 𝐵 → 𝑥 = 𝐴) | 
| 25 | 24 | fveq2d 6909 | . . . . . . . 8
⊢ (𝑥 ∈ 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐴)) | 
| 26 | 25 | adantl 481 | . . . . . . 7
⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐴)) | 
| 27 | 24 | fveq2d 6909 | . . . . . . . 8
⊢ (𝑥 ∈ 𝐵 → (𝐺‘𝑥) = (𝐺‘𝐴)) | 
| 28 | 27 | adantl 481 | . . . . . . 7
⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝐴)) | 
| 29 | 20, 26, 28 | 3eqtr4d 2786 | . . . . . 6
⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥)) | 
| 30 | 29 | adantll 714 | . . . . 5
⊢ (((𝜑 ∧ (𝐹‘𝐴) = (𝐺‘𝐴)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥)) | 
| 31 | 30 | ralrimiva 3145 | . . . 4
⊢ ((𝜑 ∧ (𝐹‘𝐴) = (𝐺‘𝐴)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) | 
| 32 | 31 | ex 412 | . . 3
⊢ (𝜑 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | 
| 33 | 19, 32 | impbid 212 | . 2
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴))) | 
| 34 | 4, 33 | bitrd 279 | 1
⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴))) |