Proof of Theorem f1o3d
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | f1o3d.2 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | 
| 2 | 1 | ralrimiva 3146 | . . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) | 
| 3 |  | eqid 2737 | . . . . . 6
⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | 
| 4 | 3 | fnmpt 6708 | . . . . 5
⊢
(∀𝑥 ∈
𝐴 𝐶 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) | 
| 5 | 2, 4 | syl 17 | . . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) | 
| 6 |  | f1o3d.1 | . . . . 5
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) | 
| 7 | 6 | fneq1d 6661 | . . . 4
⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)) | 
| 8 | 5, 7 | mpbird 257 | . . 3
⊢ (𝜑 → 𝐹 Fn 𝐴) | 
| 9 |  | f1o3d.3 | . . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) | 
| 10 | 9 | ralrimiva 3146 | . . . . 5
⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝐷 ∈ 𝐴) | 
| 11 |  | eqid 2737 | . . . . . 6
⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = (𝑦 ∈ 𝐵 ↦ 𝐷) | 
| 12 | 11 | fnmpt 6708 | . . . . 5
⊢
(∀𝑦 ∈
𝐵 𝐷 ∈ 𝐴 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) | 
| 13 | 10, 12 | syl 17 | . . . 4
⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) | 
| 14 |  | eleq1a 2836 | . . . . . . . . . . 11
⊢ (𝐶 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵)) | 
| 15 | 1, 14 | syl 17 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵)) | 
| 16 | 15 | impr 454 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → 𝑦 ∈ 𝐵) | 
| 17 |  | f1o3d.4 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) | 
| 18 | 17 | biimpar 477 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷) | 
| 19 | 18 | exp42 435 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑥 = 𝐷)))) | 
| 20 | 19 | com34 91 | . . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷)))) | 
| 21 | 20 | imp32 418 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷)) | 
| 22 | 16, 21 | jcai 516 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) | 
| 23 |  | eleq1a 2836 | . . . . . . . . . . 11
⊢ (𝐷 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴)) | 
| 24 | 9, 23 | syl 17 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴)) | 
| 25 | 24 | impr 454 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → 𝑥 ∈ 𝐴) | 
| 26 | 17 | biimpa 476 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶) | 
| 27 | 26 | exp42 435 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → 𝑦 = 𝐶)))) | 
| 28 | 27 | com23 86 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑦 = 𝐶)))) | 
| 29 | 28 | com34 91 | . . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶)))) | 
| 30 | 29 | imp32 418 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶)) | 
| 31 | 25, 30 | jcai 516 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) | 
| 32 | 22, 31 | impbida 801 | . . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) | 
| 33 | 32 | opabbidv 5209 | . . . . . 6
⊢ (𝜑 → {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)}) | 
| 34 |  | df-mpt 5226 | . . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | 
| 35 | 6, 34 | eqtrdi 2793 | . . . . . . . 8
⊢ (𝜑 → 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}) | 
| 36 | 35 | cnveqd 5886 | . . . . . . 7
⊢ (𝜑 → ◡𝐹 = ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}) | 
| 37 |  | cnvopab 6157 | . . . . . . 7
⊢ ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | 
| 38 | 36, 37 | eqtrdi 2793 | . . . . . 6
⊢ (𝜑 → ◡𝐹 = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}) | 
| 39 |  | df-mpt 5226 | . . . . . . 7
⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)} | 
| 40 | 39 | a1i 11 | . . . . . 6
⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)}) | 
| 41 | 33, 38, 40 | 3eqtr4d 2787 | . . . . 5
⊢ (𝜑 → ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)) | 
| 42 | 41 | fneq1d 6661 | . . . 4
⊢ (𝜑 → (◡𝐹 Fn 𝐵 ↔ (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵)) | 
| 43 | 13, 42 | mpbird 257 | . . 3
⊢ (𝜑 → ◡𝐹 Fn 𝐵) | 
| 44 |  | dff1o4 6856 | . . 3
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) | 
| 45 | 8, 43, 44 | sylanbrc 583 | . 2
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | 
| 46 | 45, 41 | jca 511 | 1
⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |