Theorem pmtrmvd 18345

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 18344	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷)
3		ffn 6344	. . 3 ⊢ ((𝑇‘𝑃):𝐷⟶𝐷 → (𝑇‘𝑃) Fn 𝐷)
4		fndifnfp 6761	. . 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 18341	. . . . . 6 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑧) = if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
7	6	neeq1d 3027	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (((𝑇‘𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8		iffalse 4359	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
9	8	necon1ai 2995	. . . . . . 7 ⊢ (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 → 𝑧 ∈ 𝑃)
10		iftrue 4356	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = ∪ (𝑃 ∖ {𝑧}))
11	10	adantl 474	. . . . . . . . 9 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = ∪ (𝑃 ∖ {𝑧}))
12		1onn 8066	. . . . . . . . . . 11 ⊢ 1o ∈ ω
13		simpl3 1173	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ 2o)
14		df-2o 7906	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
15	13, 14	syl6breq 4970	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ suc 1o)
16		simpr 477	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝑃)
17		dif1en 8546	. . . . . . . . . . 11 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
18	12, 15, 16, 17	mp3an2i 1445	. . . . . . . . . 10 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
19		en1uniel 8378	. . . . . . . . . 10 ⊢ ((𝑃 ∖ {𝑧}) ≈ 1o → ∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
20		eldifsni 4596	. . . . . . . . . 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 3035	. . . . . . . 8 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
23	22	ex 405	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
24	9, 23	impbid2 218	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
25	24	adantr 473	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
26	7, 25	bitrd 271	. . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (((𝑇‘𝑃)‘𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
27	26	rabbidva 3403	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧} = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃})
28		incom 4067	. . . 4 ⊢ (𝑃 ∩ 𝐷) = (𝐷 ∩ 𝑃)
29		dfin5 3838	. . . 4 ⊢ (𝐷 ∩ 𝑃) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
30	28, 29	eqtri 2803	. . 3 ⊢ (𝑃 ∩ 𝐷) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
31	27, 30	syl6eqr 2833	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧} = (𝑃 ∩ 𝐷))
32		simp2 1117	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ⊆ 𝐷)
33		df-ss 3844	. . 3 ⊢ (𝑃 ⊆ 𝐷 ↔ (𝑃 ∩ 𝐷) = 𝑃)
34	32, 33	sylib 210	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑃 ∩ 𝐷) = 𝑃)
35	5, 31, 34	3eqtrd 2819	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → dom ((𝑇‘𝑃) ∖ I ) = 𝑃)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2968  {crab 3093   ∖ cdif 3827   ∩ cin 3829   ⊆ wss 3830  ifcif 4350  {csn 4441  ∪ cuni 4712   class class class wbr 4929   I cid 5311  dom cdm 5407  suc csuc 6031   Fn wfn 6183  ⟶wf 6184  ‘cfv 6188  ωcom 7396  1_oc1o 7898  2_oc2o 7899   ≈ cen 8303  pmTrspcpmtr 18330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-om 7397  df-1o 7905  df-2o 7906  df-er 8089  df-en 8307  df-fin 8310  df-pmtr 18331

This theorem is referenced by:  pmtrfrn  18347  pmtrfb  18354  symggen  18359  pmtrdifellem2  18366  mdetralt  20921  mdetunilem7  20931