Theorem pmtrmvd 19397

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 19396	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷)
3		ffn 6670	. . 3 ⊢ ((𝑇‘𝑃):𝐷⟶𝐷 → (𝑇‘𝑃) Fn 𝐷)
4		fndifnfp 7132	. . 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 19393	. . . . . 6 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑧) = if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
7	6	neeq1d 2992	. . . . 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 2960	. . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = ∪ (𝑃 ∖ {𝑧}))
12		1onn 8578	. . . . . . . . . . 11 ⊢ 1o ∈ ω
13		simpl3 1195	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ 2o)
14		df-2o 8408	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
15	13, 14	breqtrdi 5141	. . . . . . . . . . 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 9099	. . . . . . . . . . 11 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
18	12, 15, 16, 17	mp3an2i 1469	. . . . . . . . . 10 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o)
19		en1uniel 8978	. . . . . . . . . 10 ⊢ ((𝑃 ∖ {𝑧}) ≈ 1o → ∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}))
20		eldifsni 4748	. . . . . . . . . 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 3000	. . . . . . . 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 226	. . . . . 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 3407	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧} = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃})
28		incom 4163	. . . 4 ⊢ (𝑃 ∩ 𝐷) = (𝐷 ∩ 𝑃)
29		dfin5 3911	. . . 4 ⊢ (𝐷 ∩ 𝑃) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
30	28, 29	eqtri 2760	. . 3 ⊢ (𝑃 ∩ 𝐷) = {𝑧 ∈ 𝐷 ∣ 𝑧 ∈ 𝑃}
31	27, 30	eqtr4di 2790	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → {𝑧 ∈ 𝐷 ∣ ((𝑇‘𝑃)‘𝑧) ≠ 𝑧} = (𝑃 ∩ 𝐷))
32		simp2 1138	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ⊆ 𝐷)
33		dfss2 3921	. . 3 ⊢ (𝑃 ⊆ 𝐷 ↔ (𝑃 ∩ 𝐷) = 𝑃)
34	32, 33	sylib 218	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑃 ∩ 𝐷) = 𝑃)
35	5, 31, 34	3eqtrd 2776	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → dom ((𝑇‘𝑃) ∖ I ) = 𝑃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3401   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ifcif 4481  {csn 4582  ∪ cuni 4865   class class class wbr 5100   I cid 5526  dom cdm 5632  suc csuc 6327   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  ωcom 7818  1_oc1o 8400  2_oc2o 8401   ≈ cen 8892  pmTrspcpmtr 19382

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-2o 8408  df-en 8896  df-pmtr 19383

This theorem is referenced by:  pmtrfrn  19399  pmtrfb  19406  symggen  19411  pmtrdifellem2  19418  mdetralt  22564  mdetunilem7  22574  pmtrcnel  33182  pmtrcnel2  33183