MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pmtrmvd Structured version   Visualization version   GIF version

Theorem pmtrmvd 18580
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 18579 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃):𝐷𝐷)
3 ffn 6491 . . 3 ((𝑇𝑃):𝐷𝐷 → (𝑇𝑃) Fn 𝐷)
4 fndifnfp 6919 . . 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 18576 . . . . . 6 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → ((𝑇𝑃)‘𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
76neeq1d 3049 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8 iffalse 4437 . . . . . . . 8 𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
98necon1ai 3017 . . . . . . 7 (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃)
10 iftrue 4434 . . . . . . . . . 10 (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
1110adantl 485 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = (𝑃 ∖ {𝑧}))
12 1onn 8252 . . . . . . . . . . 11 1o ∈ ω
13 simpl3 1190 . . . . . . . . . . . 12 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → 𝑃 ≈ 2o)
14 df-2o 8090 . . . . . . . . . . . 12 2o = suc 1o
1513, 14breqtrdi 5074 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → 𝑃 ≈ suc 1o)
16 simpr 488 . . . . . . . . . . 11 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → 𝑧𝑃)
17 dif1en 8739 . . . . . . . . . . 11 ((1o ∈ ω ∧ 𝑃 ≈ suc 1o𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
1812, 15, 16, 17mp3an2i 1463 . . . . . . . . . 10 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
19 en1uniel 8568 . . . . . . . . . 10 ((𝑃 ∖ {𝑧}) ≈ 1o (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
20 eldifsni 4686 . . . . . . . . . 10 ( (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2118, 19, 203syl 18 . . . . . . . . 9 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → (𝑃 ∖ {𝑧}) ≠ 𝑧)
2211, 21eqnetrd 3057 . . . . . . . 8 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝑃) → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
2322ex 416 . . . . . . 7 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑧𝑃 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
249, 23impbid2 229 . . . . . 6 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
2524adantr 484 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → (if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧𝑧𝑃))
267, 25bitrd 282 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑧𝐷) → (((𝑇𝑃)‘𝑧) ≠ 𝑧𝑧𝑃))
2726rabbidva 3428 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = {𝑧𝐷𝑧𝑃})
28 incom 4131 . . . 4 (𝑃𝐷) = (𝐷𝑃)
29 dfin5 3892 . . . 4 (𝐷𝑃) = {𝑧𝐷𝑧𝑃}
3028, 29eqtri 2824 . . 3 (𝑃𝐷) = {𝑧𝐷𝑧𝑃}
3127, 30eqtr4di 2854 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → {𝑧𝐷 ∣ ((𝑇𝑃)‘𝑧) ≠ 𝑧} = (𝑃𝐷))
32 simp2 1134 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃𝐷)
33 df-ss 3901 . . 3 (𝑃𝐷 ↔ (𝑃𝐷) = 𝑃)
3432, 33sylib 221 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑃𝐷) = 𝑃)
355, 31, 343eqtrd 2840 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → dom ((𝑇𝑃) ∖ I ) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  {crab 3113  cdif 3881  cin 3883  wss 3884  ifcif 4428  {csn 4528   cuni 4803   class class class wbr 5033   I cid 5427  dom cdm 5523  suc csuc 6165   Fn wfn 6323  wf 6324  cfv 6328  ωcom 7564  1oc1o 8082  2oc2o 8083  cen 8493  pmTrspcpmtr 18565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1o 8089  df-2o 8090  df-er 8276  df-en 8497  df-fin 8500  df-pmtr 18566
This theorem is referenced by:  pmtrfrn  18582  pmtrfb  18589  symggen  18594  pmtrdifellem2  18601  mdetralt  21217  mdetunilem7  21227  pmtrcnel  30787  pmtrcnel2  30788
  Copyright terms: Public domain W3C validator