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Theorem pmtrmvd 18047
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 18046 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃):𝐷𝐷)
3 ffn 6194 . . 3 ((𝑇𝑃):𝐷𝐷 → (𝑇𝑃) Fn 𝐷)
4 fndifnfp 6594 . . 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 18043 . . . . . 6 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → ((𝑇𝑃)‘𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
76neeq1d 2979 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8 iffalse 4227 . . . . . . . 8 𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
98necon1ai 2947 . . . . . . 7 (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃)
10 iftrue 4224 . . . . . . . . . 10 (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
1110adantl 473 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
12 1onn 7876 . . . . . . . . . . . 12 1𝑜 ∈ ω
1312a1i 11 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 1𝑜 ∈ ω)
14 simpl3 1208 . . . . . . . . . . . 12 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 𝑃 ≈ 2𝑜)
15 df-2o 7718 . . . . . . . . . . . 12 2𝑜 = suc 1𝑜
1614, 15syl6breq 4833 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 𝑃 ≈ suc 1𝑜)
17 simpr 479 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 𝑧𝑃)
18 dif1en 8346 . . . . . . . . . . 11 ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1𝑜)
1913, 16, 17, 18syl3anc 1463 . . . . . . . . . 10 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1𝑜)
20 en1uniel 8181 . . . . . . . . . 10 ((𝑃 ∖ {𝑧}) ≈ 1𝑜 (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
21 eldifsni 4454 . . . . . . . . . 10 ( (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2219, 20, 213syl 18 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2311, 22eqnetrd 2987 . . . . . . . 8 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
2423ex 449 . . . . . . 7 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
259, 24impbid2 216 . . . . . 6 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
2625adantr 472 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
277, 26bitrd 268 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧𝑧𝑃))
2827rabbidva 3316 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = {𝑧𝐷𝑧𝑃})
29 incom 3936 . . . 4 (𝑃𝐷) = (𝐷𝑃)
30 dfin5 3711 . . . 4 (𝐷𝑃) = {𝑧𝐷𝑧𝑃}
3129, 30eqtri 2770 . . 3 (𝑃𝐷) = {𝑧𝐷𝑧𝑃}
3228, 31syl6eqr 2800 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = (𝑃𝐷))
33 simp2 1129 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃𝐷)
34 df-ss 3717 . . 3 (𝑃𝐷 ↔ (𝑃𝐷) = 𝑃)
3533, 34sylib 208 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑃𝐷) = 𝑃)
365, 32, 353eqtrd 2786 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → dom ((𝑇𝑃) ∖ I ) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wne 2920  {crab 3042  cdif 3700  cin 3702  wss 3703  ifcif 4218  {csn 4309   cuni 4576   class class class wbr 4792   I cid 5161  dom cdm 5254  suc csuc 5874   Fn wfn 6032  wf 6033  cfv 6037  ωcom 7218  1𝑜c1o 7710  2𝑜c2o 7711  cen 8106  pmTrspcpmtr 18032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-om 7219  df-1o 7717  df-2o 7718  df-er 7899  df-en 8110  df-fin 8113  df-pmtr 18033
This theorem is referenced by:  pmtrfrn  18049  pmtrfb  18056  symggen  18061  pmtrdifellem2  18068  mdetralt  20587  mdetunilem7  20597
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