Theorem pmtrfinv 19447

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 2736	. . . . . . 7 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I )
4	1, 2, 3	pmtrfrn 19444	. . . . . 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 19441	. . . . 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 6695	. . . 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 6735	. . . 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 6711	. . 3 ⊢ ((𝐹 ∘ 𝐹):𝐷⟶𝐷 → (𝐹 ∘ 𝐹) Fn 𝐷)
14	10, 12, 13	3syl 18	. 2 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) Fn 𝐷)
15		fnresi 6672	. . 3 ⊢ ( I ↾ 𝐷) Fn 𝐷
16	15	a1i 11	. 2 ⊢ (𝐹 ∈ 𝑅 → ( I ↾ 𝐷) Fn 𝐷)
17	1, 2, 3	pmtrffv 19445	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) = if(𝑥 ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥))
18		iftrue 4511	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐹 ∖ I ) → if(𝑥 ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥) = ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}))
19	17, 18	sylan9eq 2791	. . . . . 6 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) = ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}))
20	19	fveq2d 6885	. . . . 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 8657	. . . . . . . . . . 11 ⊢ 1o ∈ ω
25	5	simp3d 1144	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2o)
26		df-2o 8486	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
27	25, 26	breqtrdi 5165	. . . . . . . . . . . 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 9180	. . . . . . . . . . 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 9048	. . . . . . . . . 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 3943	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ))
35	23, 34	sseldd 3964	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷)
36	1, 2, 3	pmtrffv 19445	. . . . . . 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 4511	. . . . . . . 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 10028	. . . . . . . . 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 2771	. . . . . 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 2771	. . . . 5 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘∪ (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
46	20, 45	eqtrd 2771	. . . 4 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
47	10	ffnd 6712	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑅 → 𝐹 Fn 𝐷)
48		fnelnfp 7174	. . . . . . . 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 2976	. . . . . 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 6881	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑥 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))
53		id 22	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑥 → (𝐹‘𝑥) = 𝑥)
54	52, 53	eqtrd 2771	. . . . 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 6981	. . . 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 7170	. . . 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 2781	. 2 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐹)‘𝑥) = (( I ↾ 𝐷)‘𝑥))
62	14, 16, 61	eqfnfvd 7029	1 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) = ( I ↾ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  Vcvv 3464   ∖ cdif 3928   ⊆ wss 3931  ifcif 4505  {csn 4606  ∪ cuni 4888   class class class wbr 5124   I cid 5552  dom cdm 5659  ran crn 5660   ↾ cres 5661   ∘ ccom 5663  suc csuc 6359   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  ωcom 7866  1_oc1o 8478  2_oc2o 8479   ≈ cen 8961  pmTrspcpmtr 19427

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7867  df-1o 8485  df-2o 8486  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pmtr 19428

This theorem is referenced by:  pmtrff1o  19449  pmtrfcnv  19450  symggen  19456  psgnunilem1  19479  cyc3genpmlem  33167