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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 8107 | . 2 ⊢ 2o ≠ ∅ | |
2 | pmtrrn.t | . . . . . . . 8 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
3 | pmtrrn.r | . . . . . . . 8 ⊢ 𝑅 = ran 𝑇 | |
4 | eqid 2821 | . . . . . . . 8 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I ) | |
5 | 2, 3, 4 | pmtrfrn 18580 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I )))) |
6 | 5 | simpld 497 | . . . . . 6 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o)) |
7 | 6 | simp3d 1140 | . . . . 5 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2o) |
8 | enen1 8651 | . . . . 5 ⊢ (dom (𝐹 ∖ I ) ≈ 2o → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2o ≈ ∅)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2o ≈ ∅)) |
10 | en0 8566 | . . . 4 ⊢ (dom (𝐹 ∖ I ) ≈ ∅ ↔ dom (𝐹 ∖ I ) = ∅) | |
11 | en0 8566 | . . . 4 ⊢ (2o ≈ ∅ ↔ 2o = ∅) | |
12 | 9, 10, 11 | 3bitr3g 315 | . . 3 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) = ∅ ↔ 2o = ∅)) |
13 | 12 | necon3bid 3060 | . 2 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≠ ∅ ↔ 2o ≠ ∅)) |
14 | 1, 13 | mpbiri 260 | 1 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 ∅c0 4290 class class class wbr 5058 I cid 5453 dom cdm 5549 ran crn 5550 ‘cfv 6349 2oc2o 8090 ≈ cen 8500 pmTrspcpmtr 18563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-1o 8096 df-2o 8097 df-er 8283 df-en 8504 df-fin 8507 df-pmtr 18564 |
This theorem is referenced by: psgnunilem3 18618 |
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