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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
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