Theorem pmtrfinv 19391

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 2729	. . . . . . 7 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I )
4	1, 2, 3	pmtrfrn 19388	. . . . . 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 19385	. . . . 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 6670	. . . 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 6712	. . . 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 6688	. . 3 ⊢ ((𝐹 ∘ 𝐹):𝐷⟶𝐷 → (𝐹 ∘ 𝐹) Fn 𝐷)
14	10, 12, 13	3syl 18	. 2 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) Fn 𝐷)
15		fnresi 6647	. . 3 ⊢ ( I ↾ 𝐷) Fn 𝐷
16	15	a1i 11	. 2 ⊢ (𝐹 ∈ 𝑅 → ( I ↾ 𝐷) Fn 𝐷)
17	1, 2, 3	pmtrffv 19389	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) = if(𝑥 ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥))
18		iftrue 4494	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝐹 ∖ I ) → if(𝑥 ∈ dom (𝐹 ∖ I ), ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥) = ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}))
19	17, 18	sylan9eq 2784	. . . . . 6 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) = ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}))
20	19	fveq2d 6862	. . . . 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 8604	. . . . . . . . . . 11 ⊢ 1o ∈ ω
25	5	simp3d 1144	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2o)
26		df-2o 8435	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
27	25, 26	breqtrdi 5148	. . . . . . . . . . . 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 9125	. . . . . . . . . . 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 9000	. . . . . . . . . 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 3926	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ))
35	23, 34	sseldd 3947	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → ∪ (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷)
36	1, 2, 3	pmtrffv 19389	. . . . . . 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 4494	. . . . . . . 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 9962	. . . . . . . . 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 2764	. . . . . 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 2764	. . . . 5 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘∪ (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
46	20, 45	eqtrd 2764	. . . 4 ⊢ (((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
47	10	ffnd 6689	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑅 → 𝐹 Fn 𝐷)
48		fnelnfp 7151	. . . . . . . 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 2968	. . . . . 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 6858	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑥 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))
53		id 22	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑥 → (𝐹‘𝑥) = 𝑥)
54	52, 53	eqtrd 2764	. . . . 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 6958	. . . 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 7147	. . . 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 2774	. 2 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐹)‘𝑥) = (( I ↾ 𝐷)‘𝑥))
62	14, 16, 61	eqfnfvd 7006	1 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) = ( I ↾ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3447   ∖ cdif 3911   ⊆ wss 3914  ifcif 4488  {csn 4589  ∪ cuni 4871   class class class wbr 5107   I cid 5532  dom cdm 5638  ran crn 5639   ↾ cres 5640   ∘ ccom 5642  suc csuc 6334   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  ωcom 7842  1_oc1o 8427  2_oc2o 8428   ≈ cen 8915  pmTrspcpmtr 19371

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1o 8434  df-2o 8435  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pmtr 19372

This theorem is referenced by:  pmtrff1o  19393  pmtrfcnv  19394  symggen  19400  psgnunilem1  19423  cyc3genpmlem  33108