Proof of Theorem pmtrrn2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | pmtrrn.t | . . . . . . 7
⊢ 𝑇 = (pmTrsp‘𝐷) | 
| 2 |  | pmtrrn.r | . . . . . . 7
⊢ 𝑅 = ran 𝑇 | 
| 3 |  | eqid 2736 | . . . . . . 7
⊢ dom
(𝐹 ∖ I ) = dom (𝐹 ∖ I ) | 
| 4 | 1, 2, 3 | pmtrfrn 19477 | . . . . . 6
⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧
𝐹 = (𝑇‘dom (𝐹 ∖ I )))) | 
| 5 | 4 | simpld 494 | . . . . 5
⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈
2o)) | 
| 6 | 5 | simp3d 1144 | . . . 4
⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈
2o) | 
| 7 |  | en2 9316 | . . . 4
⊢ (dom
(𝐹 ∖ I ) ≈
2o → ∃𝑥∃𝑦dom (𝐹 ∖ I ) = {𝑥, 𝑦}) | 
| 8 | 6, 7 | syl 17 | . . 3
⊢ (𝐹 ∈ 𝑅 → ∃𝑥∃𝑦dom (𝐹 ∖ I ) = {𝑥, 𝑦}) | 
| 9 | 5 | simp2d 1143 | . . . . . . 7
⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ⊆ 𝐷) | 
| 10 | 4 | simprd 495 | . . . . . . 7
⊢ (𝐹 ∈ 𝑅 → 𝐹 = (𝑇‘dom (𝐹 ∖ I ))) | 
| 11 | 9, 6, 10 | jca32 515 | . . . . . 6
⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧
𝐹 = (𝑇‘dom (𝐹 ∖ I ))))) | 
| 12 |  | sseq1 4008 | . . . . . . 7
⊢ (dom
(𝐹 ∖ I ) = {𝑥, 𝑦} → (dom (𝐹 ∖ I ) ⊆ 𝐷 ↔ {𝑥, 𝑦} ⊆ 𝐷)) | 
| 13 |  | breq1 5145 | . . . . . . . 8
⊢ (dom
(𝐹 ∖ I ) = {𝑥, 𝑦} → (dom (𝐹 ∖ I ) ≈ 2o ↔
{𝑥, 𝑦} ≈ 2o)) | 
| 14 |  | fveq2 6905 | . . . . . . . . 9
⊢ (dom
(𝐹 ∖ I ) = {𝑥, 𝑦} → (𝑇‘dom (𝐹 ∖ I )) = (𝑇‘{𝑥, 𝑦})) | 
| 15 | 14 | eqeq2d 2747 | . . . . . . . 8
⊢ (dom
(𝐹 ∖ I ) = {𝑥, 𝑦} → (𝐹 = (𝑇‘dom (𝐹 ∖ I )) ↔ 𝐹 = (𝑇‘{𝑥, 𝑦}))) | 
| 16 | 13, 15 | anbi12d 632 | . . . . . . 7
⊢ (dom
(𝐹 ∖ I ) = {𝑥, 𝑦} → ((dom (𝐹 ∖ I ) ≈ 2o ∧
𝐹 = (𝑇‘dom (𝐹 ∖ I ))) ↔ ({𝑥, 𝑦} ≈ 2o ∧ 𝐹 = (𝑇‘{𝑥, 𝑦})))) | 
| 17 | 12, 16 | anbi12d 632 | . . . . . 6
⊢ (dom
(𝐹 ∖ I ) = {𝑥, 𝑦} → ((dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧
𝐹 = (𝑇‘dom (𝐹 ∖ I )))) ↔ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ({𝑥, 𝑦} ≈ 2o ∧ 𝐹 = (𝑇‘{𝑥, 𝑦}))))) | 
| 18 | 11, 17 | syl5ibcom 245 | . . . . 5
⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) = {𝑥, 𝑦} → ({𝑥, 𝑦} ⊆ 𝐷 ∧ ({𝑥, 𝑦} ≈ 2o ∧ 𝐹 = (𝑇‘{𝑥, 𝑦}))))) | 
| 19 |  | vex 3483 | . . . . . . . 8
⊢ 𝑥 ∈ V | 
| 20 |  | vex 3483 | . . . . . . . 8
⊢ 𝑦 ∈ V | 
| 21 | 19, 20 | prss 4819 | . . . . . . 7
⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ↔ {𝑥, 𝑦} ⊆ 𝐷) | 
| 22 | 21 | bicomi 224 | . . . . . 6
⊢ ({𝑥, 𝑦} ⊆ 𝐷 ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) | 
| 23 |  | pr2ne 10045 | . . . . . . . 8
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦)) | 
| 24 | 23 | el2v 3486 | . . . . . . 7
⊢ ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦) | 
| 25 | 24 | anbi1i 624 | . . . . . 6
⊢ (({𝑥, 𝑦} ≈ 2o ∧ 𝐹 = (𝑇‘{𝑥, 𝑦})) ↔ (𝑥 ≠ 𝑦 ∧ 𝐹 = (𝑇‘{𝑥, 𝑦}))) | 
| 26 | 22, 25 | anbi12i 628 | . . . . 5
⊢ (({𝑥, 𝑦} ⊆ 𝐷 ∧ ({𝑥, 𝑦} ≈ 2o ∧ 𝐹 = (𝑇‘{𝑥, 𝑦}))) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ≠ 𝑦 ∧ 𝐹 = (𝑇‘{𝑥, 𝑦})))) | 
| 27 | 18, 26 | imbitrdi 251 | . . . 4
⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) = {𝑥, 𝑦} → ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ≠ 𝑦 ∧ 𝐹 = (𝑇‘{𝑥, 𝑦}))))) | 
| 28 | 27 | 2eximdv 1918 | . . 3
⊢ (𝐹 ∈ 𝑅 → (∃𝑥∃𝑦dom (𝐹 ∖ I ) = {𝑥, 𝑦} → ∃𝑥∃𝑦((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ≠ 𝑦 ∧ 𝐹 = (𝑇‘{𝑥, 𝑦}))))) | 
| 29 | 8, 28 | mpd 15 | . 2
⊢ (𝐹 ∈ 𝑅 → ∃𝑥∃𝑦((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ≠ 𝑦 ∧ 𝐹 = (𝑇‘{𝑥, 𝑦})))) | 
| 30 |  | r2ex 3195 | . 2
⊢
(∃𝑥 ∈
𝐷 ∃𝑦 ∈ 𝐷 (𝑥 ≠ 𝑦 ∧ 𝐹 = (𝑇‘{𝑥, 𝑦})) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ (𝑥 ≠ 𝑦 ∧ 𝐹 = (𝑇‘{𝑥, 𝑦})))) | 
| 31 | 29, 30 | sylibr 234 | 1
⊢ (𝐹 ∈ 𝑅 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 (𝑥 ≠ 𝑦 ∧ 𝐹 = (𝑇‘{𝑥, 𝑦}))) |