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Mirrors > Home > MPE Home > Th. List > pmtrfmvdn0 | Structured version Visualization version GIF version |
Description: A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
Ref | Expression |
---|---|
pmtrrn.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
pmtrrn.r | ⊢ 𝑅 = ran 𝑇 |
Ref | Expression |
---|---|
pmtrfmvdn0 | ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on0 8514 | . 2 ⊢ 2o ≠ ∅ | |
2 | pmtrrn.t | . . . . . . . 8 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
3 | pmtrrn.r | . . . . . . . 8 ⊢ 𝑅 = ran 𝑇 | |
4 | eqid 2726 | . . . . . . . 8 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I ) | |
5 | 2, 3, 4 | pmtrfrn 19458 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I )))) |
6 | 5 | simpld 493 | . . . . . 6 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o)) |
7 | 6 | simp3d 1141 | . . . . 5 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2o) |
8 | enen1 9157 | . . . . 5 ⊢ (dom (𝐹 ∖ I ) ≈ 2o → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2o ≈ ∅)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2o ≈ ∅)) |
10 | en0 9051 | . . . 4 ⊢ (dom (𝐹 ∖ I ) ≈ ∅ ↔ dom (𝐹 ∖ I ) = ∅) | |
11 | en0 9051 | . . . 4 ⊢ (2o ≈ ∅ ↔ 2o = ∅) | |
12 | 9, 10, 11 | 3bitr3g 312 | . . 3 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) = ∅ ↔ 2o = ∅)) |
13 | 12 | necon3bid 2975 | . 2 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≠ ∅ ↔ 2o ≠ ∅)) |
14 | 1, 13 | mpbiri 257 | 1 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4325 class class class wbr 5155 I cid 5581 dom cdm 5684 ran crn 5685 ‘cfv 6556 2oc2o 8492 ≈ cen 8973 pmTrspcpmtr 19441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-om 7879 df-1o 8498 df-2o 8499 df-er 8736 df-en 8977 df-pmtr 19442 |
This theorem is referenced by: psgnunilem3 19496 |
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