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Theorem pmtrmvd 19402
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 19401 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃):𝐷𝐷)
3 ffn 6716 . . 3 ((𝑇𝑃):𝐷𝐷 → (𝑇𝑃) Fn 𝐷)
4 fndifnfp 7179 . . 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 19398 . . . . . 6 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → ((𝑇𝑃)‘𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
76neeq1d 2995 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8 iffalse 4533 . . . . . . . 8 𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
98necon1ai 2963 . . . . . . 7 (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃)
10 iftrue 4530 . . . . . . . . . 10 (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
1110adantl 481 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
12 1onn 8654 . . . . . . . . . . 11 1o ∈ ω
13 simpl3 1191 . . . . . . . . . . . 12 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → 𝑃 ≈ 2o)
14 df-2o 8481 . . . . . . . . . . . 12 2o = suc 1o
1513, 14breqtrdi 5183 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → 𝑃 ≈ suc 1o)
16 simpr 484 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → 𝑧𝑃)
17 dif1ennn 9177 . . . . . . . . . . 11 ((1o ∈ ω ∧ 𝑃 ≈ suc 1o𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
1812, 15, 16, 17mp3an2i 1463 . . . . . . . . . 10 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
19 en1uniel 9044 . . . . . . . . . 10 ((𝑃 ∖ {𝑧}) ≈ 1o (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
20 eldifsni 4789 . . . . . . . . . 10 ( (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2118, 19, 203syl 18 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2211, 21eqnetrd 3003 . . . . . . . 8 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
2322ex 412 . . . . . . 7 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
249, 23impbid2 225 . . . . . 6 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
2524adantr 480 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
267, 25bitrd 279 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧𝑧𝑃))
2726rabbidva 3434 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = {𝑧𝐷𝑧𝑃})
28 incom 4197 . . . 4 (𝑃𝐷) = (𝐷𝑃)
29 dfin5 3952 . . . 4 (𝐷𝑃) = {𝑧𝐷𝑧𝑃}
3028, 29eqtri 2755 . . 3 (𝑃𝐷) = {𝑧𝐷𝑧𝑃}
3127, 30eqtr4di 2785 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = (𝑃𝐷))
32 simp2 1135 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃𝐷)
33 df-ss 3961 . . 3 (𝑃𝐷 ↔ (𝑃𝐷) = 𝑃)
3432, 33sylib 217 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑃𝐷) = 𝑃)
355, 31, 343eqtrd 2771 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → dom ((𝑇𝑃) ∖ I ) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2935  {crab 3427  cdif 3941  cin 3943  wss 3944  ifcif 4524  {csn 4624   cuni 4903   class class class wbr 5142   I cid 5569  dom cdm 5672  suc csuc 6365   Fn wfn 6537  wf 6538  cfv 6542  ωcom 7864  1oc1o 8473  2oc2o 8474  cen 8952  pmTrspcpmtr 19387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7865  df-1o 8480  df-2o 8481  df-en 8956  df-pmtr 19388
This theorem is referenced by:  pmtrfrn  19404  pmtrfb  19411  symggen  19416  pmtrdifellem2  19423  mdetralt  22497  mdetunilem7  22507  pmtrcnel  32790  pmtrcnel2  32791
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