Theorem pmtrmvd 19366

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 19365	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷)
3		ffn 6717	. . 3 ⊢ ((𝑇‘𝑃):𝐷⟶𝐷 → (𝑇‘𝑃) Fn 𝐷)
4		fndifnfp 7176	. . 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 19362	. . . . . 6 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑧) = if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
7	6	neeq1d 2999	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (((𝑇‘𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8		iffalse 4537	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
9	8	necon1ai 2967	. . . . . . 7 ⊢ (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 → 𝑧 ∈ 𝑃)
10		iftrue 4534	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = ∪ (𝑃 ∖ {𝑧}))
11	10	adantl 481	. . . . . . . . 9 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = ∪ (𝑃 ∖ {𝑧}))
12		1onn 8642	. . . . . . . . . . 11 ⊢ 1o ∈ ω
13		simpl3 1192	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ 2o)
14		df-2o 8470	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
15	13, 14	breqtrdi 5189	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ suc 1o)
16		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝑃)
17		dif1ennn 9164	. . . . . . . . . . 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 9031	. . . . . . . . . 10 ⊢ ((𝑃 ∖ {𝑧}) ≈ 1o → ∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
20		eldifsni 4793	. . . . . . . . . 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 3007	. . . . . . . 8 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
23	22	ex 412	. . . . . . 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 480	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
26	7, 25	bitrd 279	. . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (((𝑇‘𝑃)‘𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
27	26	rabbidva 3438	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧} = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃})
28		incom 4201	. . . 4 ⊢ (𝑃 ∩ 𝐷) = (𝐷 ∩ 𝑃)
29		dfin5 3956	. . . 4 ⊢ (𝐷 ∩ 𝑃) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
30	28, 29	eqtri 2759	. . 3 ⊢ (𝑃 ∩ 𝐷) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
31	27, 30	eqtr4di 2789	. 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 3965	. . 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 2775	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → dom ((𝑇‘𝑃) ∖ I ) = 𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  {crab 3431   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948  ifcif 4528  {csn 4628  ∪ cuni 4908   class class class wbr 5148   I cid 5573  dom cdm 5676  suc csuc 6366   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  ωcom 7858  1_oc1o 8462  2_oc2o 8463   ≈ cen 8939  pmTrspcpmtr 19351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7859  df-1o 8469  df-2o 8470  df-en 8943  df-pmtr 19352

This theorem is referenced by:  pmtrfrn  19368  pmtrfb  19375  symggen  19380  pmtrdifellem2  19387  mdetralt  22331  mdetunilem7  22341  pmtrcnel  32521  pmtrcnel2  32522