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Theorem pmtrmvd 19323
Description: A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Hypothesis
Ref Expression
pmtrfval.t 𝑇 = (pmTrsp‘𝐷)
Assertion
Ref Expression
pmtrmvd ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → dom ((𝑇𝑃) ∖ I ) = 𝑃)

Proof of Theorem pmtrmvd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 pmtrfval.t . . . 4 𝑇 = (pmTrsp‘𝐷)
21pmtrf 19322 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃):𝐷𝐷)
3 ffn 6717 . . 3 ((𝑇𝑃):𝐷𝐷 → (𝑇𝑃) Fn 𝐷)
4 fndifnfp 7173 . . 3 ((𝑇𝑃) Fn 𝐷 → dom ((𝑇𝑃) ∖ I ) = {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧})
52, 3, 43syl 18 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → dom ((𝑇𝑃) ∖ I ) = {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧})
61pmtrfv 19319 . . . . . 6 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → ((𝑇𝑃)‘𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
76neeq1d 3000 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8 iffalse 4537 . . . . . . . 8 𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
98necon1ai 2968 . . . . . . 7 (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃)
10 iftrue 4534 . . . . . . . . . 10 (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
1110adantl 482 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
12 1onn 8638 . . . . . . . . . . 11 1o ∈ ω
13 simpl3 1193 . . . . . . . . . . . 12 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → 𝑃 ≈ 2o)
14 df-2o 8466 . . . . . . . . . . . 12 2o = suc 1o
1513, 14breqtrdi 5189 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → 𝑃 ≈ suc 1o)
16 simpr 485 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → 𝑧𝑃)
17 dif1ennn 9160 . . . . . . . . . . 11 ((1o ∈ ω ∧ 𝑃 ≈ suc 1o𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
1812, 15, 16, 17mp3an2i 1466 . . . . . . . . . 10 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
19 en1uniel 9027 . . . . . . . . . 10 ((𝑃 ∖ {𝑧}) ≈ 1o (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
20 eldifsni 4793 . . . . . . . . . 10 ( (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2118, 19, 203syl 18 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2211, 21eqnetrd 3008 . . . . . . . 8 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
2322ex 413 . . . . . . 7 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
249, 23impbid2 225 . . . . . 6 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
2524adantr 481 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
267, 25bitrd 278 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧𝑧𝑃))
2726rabbidva 3439 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = {𝑧𝐷𝑧𝑃})
28 incom 4201 . . . 4 (𝑃𝐷) = (𝐷𝑃)
29 dfin5 3956 . . . 4 (𝐷𝑃) = {𝑧𝐷𝑧𝑃}
3028, 29eqtri 2760 . . 3 (𝑃𝐷) = {𝑧𝐷𝑧𝑃}
3127, 30eqtr4di 2790 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = (𝑃𝐷))
32 simp2 1137 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃𝐷)
33 df-ss 3965 . . 3 (𝑃𝐷 ↔ (𝑃𝐷) = 𝑃)
3432, 33sylib 217 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑃𝐷) = 𝑃)
355, 31, 343eqtrd 2776 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → dom ((𝑇𝑃) ∖ I ) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  {crab 3432  cdif 3945  cin 3947  wss 3948  ifcif 4528  {csn 4628   cuni 4908   class class class wbr 5148   I cid 5573  dom cdm 5676  suc csuc 6366   Fn wfn 6538  wf 6539  cfv 6543  ωcom 7854  1oc1o 8458  2oc2o 8459  cen 8935  pmTrspcpmtr 19308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-2o 8466  df-en 8939  df-pmtr 19309
This theorem is referenced by:  pmtrfrn  19325  pmtrfb  19332  symggen  19337  pmtrdifellem2  19344  mdetralt  22109  mdetunilem7  22119  pmtrcnel  32245  pmtrcnel2  32246
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