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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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