Theorem pmtrfconj 19074

Description: Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Hypotheses

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

Assertion

Ref	Expression
pmtrfconj	⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∈ 𝑅)

Proof of Theorem pmtrfconj

Step	Hyp	Ref	Expression
1		pmtrrn.t	. . . . 5 ⊢ 𝑇 = (pmTrsp‘𝐷)
2		pmtrrn.r	. . . . 5 ⊢ 𝑅 = ran 𝑇
3	1, 2	pmtrfb 19073	. . . 4 ⊢ (𝐹 ∈ 𝑅 ↔ (𝐷 ∈ V ∧ 𝐹:𝐷–1-1-onto→𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o))
4	3	simp1bi 1144	. . 3 ⊢ (𝐹 ∈ 𝑅 → 𝐷 ∈ V)
5	4	adantr 481	. 2 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐷 ∈ V)
6		simpr 485	. . . 4 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐺:𝐷–1-1-onto→𝐷)
7	1, 2	pmtrff1o 19071	. . . . 5 ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷–1-1-onto→𝐷)
8	7	adantr 481	. . . 4 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐹:𝐷–1-1-onto→𝐷)
9		f1oco 6739	. . . 4 ⊢ ((𝐺:𝐷–1-1-onto→𝐷 ∧ 𝐹:𝐷–1-1-onto→𝐷) → (𝐺 ∘ 𝐹):𝐷–1-1-onto→𝐷)
10	6, 8, 9	syl2anc 584	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → (𝐺 ∘ 𝐹):𝐷–1-1-onto→𝐷)
11		f1ocnv 6728	. . . 4 ⊢ (𝐺:𝐷–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐷)
12	11	adantl 482	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ◡𝐺:𝐷–1-1-onto→𝐷)
13		f1oco 6739	. . 3 ⊢ (((𝐺 ∘ 𝐹):𝐷–1-1-onto→𝐷 ∧ ◡𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐷–1-1-onto→𝐷)
14	10, 12, 13	syl2anc 584	. 2 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐷–1-1-onto→𝐷)
15		f1of 6716	. . . . . . 7 ⊢ (𝐹:𝐷–1-1-onto→𝐷 → 𝐹:𝐷⟶𝐷)
16	7, 15	syl 17	. . . . . 6 ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷⟶𝐷)
17	16	adantr 481	. . . . 5 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐹:𝐷⟶𝐷)
18		f1omvdconj 19054	. . . . 5 ⊢ ((𝐹:𝐷⟶𝐷 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I )))
19	17, 6, 18	syl2anc 584	. . . 4 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I )))
20		f1of1 6715	. . . . . 6 ⊢ (𝐺:𝐷–1-1-onto→𝐷 → 𝐺:𝐷–1-1→𝐷)
21	20	adantl 482	. . . . 5 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → 𝐺:𝐷–1-1→𝐷)
22		difss 4066	. . . . . . 7 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
23		dmss 5811	. . . . . . 7 ⊢ ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
24	22, 23	ax-mp 5	. . . . . 6 ⊢ dom (𝐹 ∖ I ) ⊆ dom 𝐹
25	24, 17	fssdm 6620	. . . . 5 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ⊆ 𝐷)
26	5, 25	ssexd 5248	. . . . 5 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ∈ V)
27		f1imaeng 8800	. . . . 5 ⊢ ((𝐺:𝐷–1-1→𝐷 ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ∈ V) → (𝐺 “ dom (𝐹 ∖ I )) ≈ dom (𝐹 ∖ I ))
28	21, 25, 26, 27	syl3anc 1370	. . . 4 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → (𝐺 “ dom (𝐹 ∖ I )) ≈ dom (𝐹 ∖ I ))
29	19, 28	eqbrtrd 5096	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ dom (𝐹 ∖ I ))
30	3	simp3bi 1146	. . . 4 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2o)
31	30	adantr 481	. . 3 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (𝐹 ∖ I ) ≈ 2o)
32		entr 8792	. . 3 ⊢ ((dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2o) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ 2o)
33	29, 31, 32	syl2anc 584	. 2 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ 2o)
34	1, 2	pmtrfb 19073	. 2 ⊢ (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∈ 𝑅 ↔ (𝐷 ∈ V ∧ ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐷–1-1-onto→𝐷 ∧ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ≈ 2o))
35	5, 14, 33, 34	syl3anbrc 1342	1 ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∈ 𝑅)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887   class class class wbr 5074   I cid 5488  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592   ∘ ccom 5593  ⟶wf 6429  –1-1→wf1 6430  –1-1-onto→wf1o 6432  ‘cfv 6433  2_oc2o 8291   ≈ cen 8730  pmTrspcpmtr 19049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-2o 8298  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pmtr 19050

This theorem is referenced by:  psgnunilem1  19101