Theorem pmtrmvd 19064

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	⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → dom ((𝑇‘𝑃) ∖ I ) = 𝑃)

Proof of Theorem pmtrmvd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmtrfval.t	. . . 4 ⊢ 𝑇 = (pmTrsp‘𝐷)
2	1	pmtrf 19063	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷)
3		ffn 6600	. . 3 ⊢ ((𝑇‘𝑃):𝐷⟶𝐷 → (𝑇‘𝑃) Fn 𝐷)
4		fndifnfp 7048	. . 3 ⊢ ((𝑇‘𝑃) Fn 𝐷 → dom ((𝑇‘𝑃) ∖ I ) = {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧})
5	2, 3, 4	3syl 18	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → dom ((𝑇‘𝑃) ∖ I ) = {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧})
6	1	pmtrfv 19060	. . . . . 6 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑧) = if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
7	6	neeq1d 3003	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (((𝑇‘𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8		iffalse 4468	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
9	8	necon1ai 2971	. . . . . . 7 ⊢ (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 → 𝑧 ∈ 𝑃)
10		iftrue 4465	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = ∪ (𝑃 ∖ {𝑧}))
11	10	adantl 482	. . . . . . . . 9 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = ∪ (𝑃 ∖ {𝑧}))
12		1onn 8470	. . . . . . . . . . 11 ⊢ 1o ∈ ω
13		simpl3 1192	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ 2o)
14		df-2o 8298	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
15	13, 14	breqtrdi 5115	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ suc 1o)
16		simpr 485	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝑃)
17		dif1en 8945	. . . . . . . . . . 11 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
18	12, 15, 16, 17	mp3an2i 1465	. . . . . . . . . 10 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
19		en1uniel 8818	. . . . . . . . . 10 ⊢ ((𝑃 ∖ {𝑧}) ≈ 1o → ∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
20		eldifsni 4723	. . . . . . . . . 10 ⊢ (∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}) → ∪ (𝑃 ∖ {𝑧}) ≠ 𝑧)
21	18, 19, 20	3syl 18	. . . . . . . . 9 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → ∪ (𝑃 ∖ {𝑧}) ≠ 𝑧)
22	11, 21	eqnetrd 3011	. . . . . . . 8 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
23	22	ex 413	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
24	9, 23	impbid2 225	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
25	24	adantr 481	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
26	7, 25	bitrd 278	. . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (((𝑇‘𝑃)‘𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
27	26	rabbidva 3413	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧} = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃})
28		incom 4135	. . . 4 ⊢ (𝑃 ∩ 𝐷) = (𝐷 ∩ 𝑃)
29		dfin5 3895	. . . 4 ⊢ (𝐷 ∩ 𝑃) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
30	28, 29	eqtri 2766	. . 3 ⊢ (𝑃 ∩ 𝐷) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
31	27, 30	eqtr4di 2796	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧} = (𝑃 ∩ 𝐷))
32		simp2 1136	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ⊆ 𝐷)
33		df-ss 3904	. . 3 ⊢ (𝑃 ⊆ 𝐷 ↔ (𝑃 ∩ 𝐷) = 𝑃)
34	32, 33	sylib 217	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑃 ∩ 𝐷) = 𝑃)
35	5, 31, 34	3eqtrd 2782	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → dom ((𝑇‘𝑃) ∖ I ) = 𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ifcif 4459  {csn 4561  ∪ cuni 4839   class class class wbr 5074   I cid 5488  dom cdm 5589  suc csuc 6268   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  ωcom 7712  1_oc1o 8290  2_oc2o 8291   ≈ cen 8730  pmTrspcpmtr 19049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-2o 8298  df-en 8734  df-pmtr 19050

This theorem is referenced by:  pmtrfrn  19066  pmtrfb  19073  symggen  19078  pmtrdifellem2  19085  mdetralt  21757  mdetunilem7  21767  pmtrcnel  31358  pmtrcnel2  31359