Theorem pmtrfinv 19377

Description: A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Hypotheses

Ref	Expression
pmtrrn.t	⊢ 𝑇 = (pmTrsp‘𝐷)
pmtrrn.r	⊢ 𝑅 = ran 𝑇

Assertion

Ref	Expression
pmtrfinv	⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) = ( I ↾ 𝐷))

Proof of Theorem pmtrfinv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmtrrn.t	. . . . . . 7 ⊢ 𝑇 = (pmTrsp‘𝐷)
2		pmtrrn.r	. . . . . . 7 ⊢ 𝑅 = ran 𝑇
3		eqid 2733	. . . . . . 7 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I )
4	1, 2, 3	pmtrfrn 19374	. . . . . 6 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I ))))
5	4	simpld 494	. . . . 5 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o))
6	1	pmtrf 19371	. . . . 5 ⊢ ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) → (𝑇‘dom (𝐹 ∖ I )):𝐷⟶𝐷)
7	5, 6	syl 17	. . . 4 ⊢ (𝐹 ∈ 𝑅 → (𝑇‘dom (𝐹 ∖ I )):𝐷⟶𝐷)
8	4	simprd 495	. . . . 5 ⊢ (𝐹 ∈ 𝑅 → 𝐹 = (𝑇‘dom (𝐹 ∖ I )))
9	8	feq1d 6640	. . . 4 ⊢ (𝐹 ∈ 𝑅 → (𝐹:𝐷⟶𝐷 ↔ (𝑇‘dom (𝐹 ∖ I )):𝐷⟶𝐷))
10	7, 9	mpbird 257	. . 3 ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷⟶𝐷)
11		fco 6682	. . . 4 ⊢ ((𝐹:𝐷⟶𝐷 ∧ 𝐹:𝐷⟶𝐷) → (𝐹 ∘ 𝐹):𝐷⟶𝐷)
12	11	anidms 566	. . 3 ⊢ (𝐹:𝐷⟶𝐷 → (𝐹 ∘ 𝐹):𝐷⟶𝐷)
13		ffn 6658	. . 3 ⊢ ((𝐹 ∘ 𝐹):𝐷⟶𝐷 → (𝐹 ∘ 𝐹) Fn 𝐷)
14	10, 12, 13	3syl 18	. 2 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) Fn 𝐷)
15		fnresi 6617	. . 3 ⊢ ( I ↾ 𝐷) Fn 𝐷
16	15	a1i 11	. 2 ⊢ (𝐹 ∈ 𝑅 → ( I ↾ 𝐷) Fn 𝐷)
17	1, 2, 3	pmtrffv 19375	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) = if(𝑥 ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥))
18		iftrue 4482	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐹 ∖ I ) → if(𝑥 ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥) = ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}))
19	17, 18	sylan9eq 2788	. . . . . 6 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) = ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}))
20	19	fveq2d 6834	. . . . 5 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘∪ (dom (𝐹 ∖ I ) ∖ {𝑥})))
21		simpll 766	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝐹 ∈ 𝑅)
22	5	simp2d 1143	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ⊆ 𝐷)
23	22	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ⊆ 𝐷)
24		1onn 8563	. . . . . . . . . . 11 ⊢ 1o ∈ ω
25	5	simp3d 1144	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2o)
26		df-2o 8394	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
27	25, 26	breqtrdi 5136	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ suc 1o)
28	27	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ≈ suc 1o)
29		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝑥 ∈ dom (𝐹 ∖ I ))
30		dif1ennn 9081	. . . . . . . . . . 11 ⊢ ((1o ∈ ω ∧ dom (𝐹 ∖ I ) ≈ suc 1o ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o)
31	24, 28, 29, 30	mp3an2i 1468	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o)
32		en1uniel 8960	. . . . . . . . . 10 ⊢ ((dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
33	31, 32	syl 17	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
34	33	eldifad 3910	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ))
35	23, 34	sseldd 3931	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷)
36	1, 2, 3	pmtrffv 19375	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑅 ∧ ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷) → (𝐹‘∪ (dom (𝐹 ∖ I ) ∖ {𝑥})) = if(∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥})))
37	21, 35, 36	syl2anc 584	. . . . . 6 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘∪ (dom (𝐹 ∖ I ) ∖ {𝑥})) = if(∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥})))
38		iftrue 4482	. . . . . . . 8 ⊢ (∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ) → if(∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥})) = ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}))
39	34, 38	syl 17	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → if(∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥})) = ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}))
40	25	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → dom (𝐹 ∖ I ) ≈ 2o)
41		en2other2 9909	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2o) → ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
42	41	ancoms 458	. . . . . . . 8 ⊢ ((dom (𝐹 ∖ I ) ≈ 2o ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
43	40, 42	sylan 580	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
44	39, 43	eqtrd 2768	. . . . . 6 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → if(∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
45	37, 44	eqtrd 2768	. . . . 5 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘∪ (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
46	20, 45	eqtrd 2768	. . . 4 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
47	10	ffnd 6659	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑅 → 𝐹 Fn 𝐷)
48		fnelnfp 7119	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
49	47, 48	sylan 580	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
50	49	necon2bbid 2972	. . . . . 6 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑥) = 𝑥 ↔ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
51	50	biimpar 477	. . . . 5 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) = 𝑥)
52		fveq2 6830	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑥 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))
53		id 22	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑥 → (𝐹‘𝑥) = 𝑥)
54	52, 53	eqtrd 2768	. . . . 5 ⊢ ((𝐹‘𝑥) = 𝑥 → (𝐹‘(𝐹‘𝑥)) = 𝑥)
55	51, 54	syl 17	. . . 4 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
56	46, 55	pm2.61dan 812	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
57		fvco2 6927	. . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐹)‘𝑥) = (𝐹‘(𝐹‘𝑥)))
58	47, 57	sylan 580	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐹)‘𝑥) = (𝐹‘(𝐹‘𝑥)))
59		fvresi 7115	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (( I ↾ 𝐷)‘𝑥) = 𝑥)
60	59	adantl 481	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (( I ↾ 𝐷)‘𝑥) = 𝑥)
61	56, 58, 60	3eqtr4d 2778	. 2 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐹)‘𝑥) = (( I ↾ 𝐷)‘𝑥))
62	14, 16, 61	eqfnfvd 6975	1 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) = ( I ↾ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  Vcvv 3437   ∖ cdif 3895   ⊆ wss 3898  ifcif 4476  {csn 4577  ∪ cuni 4860   class class class wbr 5095   I cid 5515  dom cdm 5621  ran crn 5622   ↾ cres 5623   ∘ ccom 5625  suc csuc 6315   Fn wfn 6483  ⟶wf 6484  ‘cfv 6488  ωcom 7804  1_oc1o 8386  2_oc2o 8387   ≈ cen 8874  pmTrspcpmtr 19357

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-om 7805  df-1o 8393  df-2o 8394  df-er 8630  df-en 8878  df-dom 8879  df-sdom 8880  df-fin 8881  df-pmtr 19358

This theorem is referenced by:  pmtrff1o  19379  pmtrfcnv  19380  symggen  19386  psgnunilem1  19409  cyc3genpmlem  33129