Theorem pmtrmvd 19671

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 19670	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷)
3		ffn 6709	. . 3 ⊢ ((𝑇‘𝑃):𝐷⟶𝐷 → (𝑇‘𝑃) Fn 𝐷)
4		fndifnfp 7181	. . 3 ⊢ ((𝑇‘𝑃) Fn 𝐷 → dom ((𝑇‘𝑃) ∖ I ) = {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧})
5	2, 3, 4	3syl 19	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → dom ((𝑇‘𝑃) ∖ I ) = {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧})
6	1	pmtrfv 19667	. . . . . 6 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑧) = if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
7	6	neeq1d 3014	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (((𝑇‘𝑃)‘𝑧) ≠ 𝑧 ↔ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
8		iffalse 4490	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = 𝑧)
9	8	necon1ai 2982	. . . . . . 7 ⊢ (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 → 𝑧 ∈ 𝑃)
10		iftrue 4487	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = ∪ (𝑃 ∖ {𝑧}))
11	10	adantl 487	. . . . . . . . 9 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = ∪ (𝑃 ∖ {𝑧}))
12		1onn 8650	. . . . . . . . . . 11 ⊢ 1o ∈ ω
13		simpl3 1212	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ 2o)
14		df-2o 8478	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
15	13, 14	breqtrdi 5145	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ suc 1o)
16		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝑃)
17		dif1ennn 9179	. . . . . . . . . . 11 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
18	12, 15, 16, 17	mp3an2i 1495	. . . . . . . . . 10 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
19		en1uniel 9058	. . . . . . . . . 10 ⊢ ((𝑃 ∖ {𝑧}) ≈ 1o → ∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
20		eldifsni 4752	. . . . . . . . . 10 ⊢ (∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}) → ∪ (𝑃 ∖ {𝑧}) ≠ 𝑧)
21	18, 19, 20	3syl 19	. . . . . . . . 9 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → ∪ (𝑃 ∖ {𝑧}) ≠ 𝑧)
22	11, 21	eqnetrd 3022	. . . . . . . 8 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧)
23	22	ex 418	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑧 ∈ 𝑃 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧))
24	9, 23	impbid2 229	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
25	24	adantr 486	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
26	7, 25	bitrd 282	. . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → (((𝑇‘𝑃)‘𝑧) ≠ 𝑧 ↔ 𝑧 ∈ 𝑃))
27	26	rabbidva 3418	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧} = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃})
28		incom 4106	. . . 4 ⊢ (𝑃 ∩ 𝐷) = (𝐷 ∩ 𝑃)
29		dfin5 4092	. . . 4 ⊢ (𝐷 ∩ 𝑃) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
30	28, 29	eqtri 2783	. . 3 ⊢ (𝑃 ∩ 𝐷) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
31	27, 30	eqtr4di 2813	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧} = (𝑃 ∩ 𝐷))
32		simp2 1155	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ⊆ 𝐷)
33		dfss2 4094	. . 3 ⊢ (𝑃 ⊆ 𝐷 ↔ (𝑃 ∩ 𝐷) = 𝑃)
34	32, 33	sylib 221	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑃 ∩ 𝐷) = 𝑃)
35	5, 31, 34	3eqtrd 2799	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → dom ((𝑇‘𝑃) ∖ I ) = 𝑃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  {crab 3412   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ifcif 4481  {csn 4583  ∪ cuni 4866   class class class wbr 5102   I cid 5544  dom cdm 5650  suc csuc 6364   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  ωcom 7877  1_oc1o 8470  2_oc2o 8471   ≈ cen 8971  pmTrspcpmtr 19656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8477  df-2o 8478  df-en 8975  df-pmtr 19657

This theorem is used by:  pmtrfrn  19673  pmtrfb  19680  symggen  19685  pmtrdifellem2  19692  mdetralt  22924  mdetunilem7  22934  pmtrcnel  33693  pmtrcnel2  33694