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Theorem pmtrmvd 17800
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 17799 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃):𝐷𝐷)
3 ffn 6004 . . 3 ((𝑇𝑃):𝐷𝐷 → (𝑇𝑃) Fn 𝐷)
4 fndifnfp 6399 . . 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 17796 . . . . . 6 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → ((𝑇𝑃)‘𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
76neeq1d 2849 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8 iffalse 4069 . . . . . . . 8 𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
98necon1ai 2817 . . . . . . 7 (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃)
10 iftrue 4066 . . . . . . . . . 10 (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
1110adantl 482 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
12 1onn 7667 . . . . . . . . . . . 12 1𝑜 ∈ ω
1312a1i 11 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 1𝑜 ∈ ω)
14 simpl3 1064 . . . . . . . . . . . 12 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 𝑃 ≈ 2𝑜)
15 df-2o 7509 . . . . . . . . . . . 12 2𝑜 = suc 1𝑜
1614, 15syl6breq 4656 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 𝑃 ≈ suc 1𝑜)
17 simpr 477 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 𝑧𝑃)
18 dif1en 8140 . . . . . . . . . . 11 ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1𝑜)
1913, 16, 17, 18syl3anc 1323 . . . . . . . . . 10 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1𝑜)
20 en1uniel 7975 . . . . . . . . . 10 ((𝑃 ∖ {𝑧}) ≈ 1𝑜 (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
21 eldifsni 4291 . . . . . . . . . 10 ( (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2219, 20, 213syl 18 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2311, 22eqnetrd 2857 . . . . . . . 8 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
2423ex 450 . . . . . . 7 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
259, 24impbid2 216 . . . . . 6 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
2625adantr 481 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
277, 26bitrd 268 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧𝑧𝑃))
2827rabbidva 3176 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = {𝑧𝐷𝑧𝑃})
29 incom 3785 . . . 4 (𝑃𝐷) = (𝐷𝑃)
30 dfin5 3564 . . . 4 (𝐷𝑃) = {𝑧𝐷𝑧𝑃}
3129, 30eqtri 2643 . . 3 (𝑃𝐷) = {𝑧𝐷𝑧𝑃}
3228, 31syl6eqr 2673 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = (𝑃𝐷))
33 simp2 1060 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃𝐷)
34 df-ss 3570 . . 3 (𝑃𝐷 ↔ (𝑃𝐷) = 𝑃)
3533, 34sylib 208 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑃𝐷) = 𝑃)
365, 32, 353eqtrd 2659 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → dom ((𝑇𝑃) ∖ I ) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  {crab 2911  cdif 3553  cin 3555  wss 3556  ifcif 4060  {csn 4150   cuni 4404   class class class wbr 4615   I cid 4986  dom cdm 5076  suc csuc 5686   Fn wfn 5844  wf 5845  cfv 5849  ωcom 7015  1𝑜c1o 7501  2𝑜c2o 7502  cen 7899  pmTrspcpmtr 17785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-om 7016  df-1o 7508  df-2o 7509  df-er 7690  df-en 7903  df-fin 7906  df-pmtr 17786
This theorem is referenced by:  pmtrfrn  17802  pmtrfb  17809  symggen  17814  pmtrdifellem2  17821  mdetralt  20336  mdetunilem7  20346
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