Theorem pmtrfinv 19370

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 2730	. . . . . . 7 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I )
4	1, 2, 3	pmtrfrn 19367	. . . . . 6 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I ))))
5	4	simpld 493	. . . . 5 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o))
6	1	pmtrf 19364	. . . . 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 494	. . . . 5 ⊢ (𝐹 ∈ 𝑅 → 𝐹 = (𝑇‘dom (𝐹 ∖ I )))
9	8	feq1d 6701	. . . 4 ⊢ (𝐹 ∈ 𝑅 → (𝐹:𝐷⟶𝐷 ↔ (𝑇‘dom (𝐹 ∖ I )):𝐷⟶𝐷))
10	7, 9	mpbird 256	. . 3 ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷⟶𝐷)
11		fco 6740	. . . 4 ⊢ ((𝐹:𝐷⟶𝐷 ∧ 𝐹:𝐷⟶𝐷) → (𝐹 ∘ 𝐹):𝐷⟶𝐷)
12	11	anidms 565	. . 3 ⊢ (𝐹:𝐷⟶𝐷 → (𝐹 ∘ 𝐹):𝐷⟶𝐷)
13		ffn 6716	. . 3 ⊢ ((𝐹 ∘ 𝐹):𝐷⟶𝐷 → (𝐹 ∘ 𝐹) Fn 𝐷)
14	10, 12, 13	3syl 18	. 2 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) Fn 𝐷)
15		fnresi 6678	. . 3 ⊢ ( I ↾ 𝐷) Fn 𝐷
16	15	a1i 11	. 2 ⊢ (𝐹 ∈ 𝑅 → ( I ↾ 𝐷) Fn 𝐷)
17	1, 2, 3	pmtrffv 19368	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) = if(𝑥 ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥))
18		iftrue 4533	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐹 ∖ I ) → if(𝑥 ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥) = ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}))
19	17, 18	sylan9eq 2790	. . . . . 6 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) = ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}))
20	19	fveq2d 6894	. . . . 5 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘∪ (dom (𝐹 ∖ I ) ∖ {𝑥})))
21		simpll 763	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝐹 ∈ 𝑅)
22	5	simp2d 1141	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ⊆ 𝐷)
23	22	ad2antrr 722	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ⊆ 𝐷)
24		1onn 8641	. . . . . . . . . . 11 ⊢ 1o ∈ ω
25	5	simp3d 1142	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2o)
26		df-2o 8469	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
27	25, 26	breqtrdi 5188	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ suc 1o)
28	27	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ≈ suc 1o)
29		simpr 483	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝑥 ∈ dom (𝐹 ∖ I ))
30		dif1ennn 9163	. . . . . . . . . . 11 ⊢ ((1o ∈ ω ∧ dom (𝐹 ∖ I ) ≈ suc 1o ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o)
31	24, 28, 29, 30	mp3an2i 1464	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o)
32		en1uniel 9030	. . . . . . . . . 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 3959	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ))
35	23, 34	sseldd 3982	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷)
36	1, 2, 3	pmtrffv 19368	. . . . . . 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 582	. . . . . 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 4533	. . . . . . . 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 479	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → dom (𝐹 ∖ I ) ≈ 2o)
41		en2other2 10006	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2o) → ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
42	41	ancoms 457	. . . . . . . 8 ⊢ ((dom (𝐹 ∖ I ) ≈ 2o ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
43	40, 42	sylan 578	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {∪ (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
44	39, 43	eqtrd 2770	. . . . . 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 2770	. . . . 5 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘∪ (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
46	20, 45	eqtrd 2770	. . . 4 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
47	10	ffnd 6717	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑅 → 𝐹 Fn 𝐷)
48		fnelnfp 7176	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
49	47, 48	sylan 578	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
50	49	necon2bbid 2982	. . . . . 6 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑥) = 𝑥 ↔ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
51	50	biimpar 476	. . . . 5 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) = 𝑥)
52		fveq2 6890	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑥 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))
53		id 22	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑥 → (𝐹‘𝑥) = 𝑥)
54	52, 53	eqtrd 2770	. . . . 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 809	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
57		fvco2 6987	. . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐹)‘𝑥) = (𝐹‘(𝐹‘𝑥)))
58	47, 57	sylan 578	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐹)‘𝑥) = (𝐹‘(𝐹‘𝑥)))
59		fvresi 7172	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (( I ↾ 𝐷)‘𝑥) = 𝑥)
60	59	adantl 480	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (( I ↾ 𝐷)‘𝑥) = 𝑥)
61	56, 58, 60	3eqtr4d 2780	. 2 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐹)‘𝑥) = (( I ↾ 𝐷)‘𝑥))
62	14, 16, 61	eqfnfvd 7034	1 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) = ( I ↾ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  Vcvv 3472   ∖ cdif 3944   ⊆ wss 3947  ifcif 4527  {csn 4627  ∪ cuni 4907   class class class wbr 5147   I cid 5572  dom cdm 5675  ran crn 5676   ↾ cres 5677   ∘ ccom 5679  suc csuc 6365   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542  ωcom 7857  1_oc1o 8461  2_oc2o 8462   ≈ cen 8938  pmTrspcpmtr 19350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-2o 8469  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pmtr 19351

This theorem is referenced by:  pmtrff1o  19372  pmtrfcnv  19373  symggen  19379  psgnunilem1  19402  cyc3genpmlem  32580