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Theorem pmtrmvd 18076
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 18075 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃):𝐷𝐷)
3 ffn 6183 . . 3 ((𝑇𝑃):𝐷𝐷 → (𝑇𝑃) Fn 𝐷)
4 fndifnfp 6584 . . 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 18072 . . . . . 6 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → ((𝑇𝑃)‘𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
76neeq1d 3002 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8 iffalse 4234 . . . . . . . 8 𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
98necon1ai 2970 . . . . . . 7 (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃)
10 iftrue 4231 . . . . . . . . . 10 (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
1110adantl 467 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
12 1onn 7871 . . . . . . . . . . . 12 1𝑜 ∈ ω
1312a1i 11 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 1𝑜 ∈ ω)
14 simpl3 1231 . . . . . . . . . . . 12 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 𝑃 ≈ 2𝑜)
15 df-2o 7712 . . . . . . . . . . . 12 2𝑜 = suc 1𝑜
1614, 15syl6breq 4827 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 𝑃 ≈ suc 1𝑜)
17 simpr 471 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → 𝑧𝑃)
18 dif1en 8347 . . . . . . . . . . 11 ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1𝑜)
1913, 16, 17, 18syl3anc 1476 . . . . . . . . . 10 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1𝑜)
20 en1uniel 8179 . . . . . . . . . 10 ((𝑃 ∖ {𝑧}) ≈ 1𝑜 (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
21 eldifsni 4457 . . . . . . . . . 10 ( (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2219, 20, 213syl 18 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2311, 22eqnetrd 3010 . . . . . . . 8 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
2423ex 397 . . . . . . 7 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
259, 24impbid2 216 . . . . . 6 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
2625adantr 466 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
277, 26bitrd 268 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧𝑧𝑃))
2827rabbidva 3338 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = {𝑧𝐷𝑧𝑃})
29 incom 3956 . . . 4 (𝑃𝐷) = (𝐷𝑃)
30 dfin5 3731 . . . 4 (𝐷𝑃) = {𝑧𝐷𝑧𝑃}
3129, 30eqtri 2793 . . 3 (𝑃𝐷) = {𝑧𝐷𝑧𝑃}
3228, 31syl6eqr 2823 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = (𝑃𝐷))
33 simp2 1131 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃𝐷)
34 df-ss 3737 . . 3 (𝑃𝐷 ↔ (𝑃𝐷) = 𝑃)
3533, 34sylib 208 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑃𝐷) = 𝑃)
365, 32, 353eqtrd 2809 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → dom ((𝑇𝑃) ∖ I ) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  {crab 3065  cdif 3720  cin 3722  wss 3723  ifcif 4225  {csn 4316   cuni 4574   class class class wbr 4786   I cid 5156  dom cdm 5249  suc csuc 5866   Fn wfn 6024  wf 6025  cfv 6029  ωcom 7210  1𝑜c1o 7704  2𝑜c2o 7705  cen 8104  pmTrspcpmtr 18061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-om 7211  df-1o 7711  df-2o 7712  df-er 7894  df-en 8108  df-fin 8111  df-pmtr 18062
This theorem is referenced by:  pmtrfrn  18078  pmtrfb  18085  symggen  18090  pmtrdifellem2  18097  mdetralt  20625  mdetunilem7  20635
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