Theorem mvdco 19477

Description: Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
mvdco	⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))

Proof of Theorem mvdco

Step	Hyp	Ref	Expression
1		inundif 4484	. . . . . . . 8 ⊢ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I )) = 𝐺
2	1	coeq2i 5873	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = (𝐹 ∘ 𝐺)
3		coundi 6268	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
4	2, 3	eqtr3i 2764	. . . . . 6 ⊢ (𝐹 ∘ 𝐺) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
5	4	difeq1i 4131	. . . . 5 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I )
6		difundir 4296	. . . . 5 ⊢ (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
7	5, 6	eqtri 2762	. . . 4 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
8	7	dmeqi 5917	. . 3 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
9		dmun 5923	. . 3 ⊢ dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
10	8, 9	eqtri 2762	. 2 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
11		inss2 4245	. . . . . 6 ⊢ (𝐺 ∩ I ) ⊆ I
12		coss2 5869	. . . . . 6 ⊢ ((𝐺 ∩ I ) ⊆ I → (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I ))
13	11, 12	ax-mp 5	. . . . 5 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I )
14		cocnvcnv1 6278	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = (𝐹 ∘ I )
15		relcnv 6124	. . . . . . . 8 ⊢ Rel ◡◡𝐹
16		coi1 6283	. . . . . . . 8 ⊢ (Rel ◡◡𝐹 → (◡◡𝐹 ∘ I ) = ◡◡𝐹)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = ◡◡𝐹
18	14, 17	eqtr3i 2764	. . . . . 6 ⊢ (𝐹 ∘ I ) = ◡◡𝐹
19		cnvcnvss 6215	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
20	18, 19	eqsstri 4029	. . . . 5 ⊢ (𝐹 ∘ I ) ⊆ 𝐹
21	13, 20	sstri 4004	. . . 4 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹
22		ssdif 4153	. . . 4 ⊢ ((𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹 → ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ))
23		dmss 5915	. . . 4 ⊢ (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ) → dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ dom (𝐹 ∖ I ))
24	21, 22, 23	mp2b 10	. . 3 ⊢ dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ dom (𝐹 ∖ I )
25		difss 4145	. . . . 5 ⊢ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I ))
26		dmss 5915	. . . . 5 ⊢ (((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I )) → dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐹 ∘ (𝐺 ∖ I )))
27	25, 26	ax-mp 5	. . . 4 ⊢ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐹 ∘ (𝐺 ∖ I ))
28		dmcoss 5987	. . . 4 ⊢ dom (𝐹 ∘ (𝐺 ∖ I )) ⊆ dom (𝐺 ∖ I )
29	27, 28	sstri 4004	. . 3 ⊢ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )
30		unss12 4197	. . 3 ⊢ ((dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ dom (𝐹 ∖ I ) ∧ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )) → (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I )))
31	24, 29, 30	mp2an 692	. 2 ⊢ (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
32	10, 31	eqsstri 4029	1 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))

Syntax hints:   = wceq 1536   ∖ cdif 3959   ∪ cun 3960   ∩ cin 3961   ⊆ wss 3962   I cid 5581  ^◡ccnv 5687  dom cdm 5688   ∘ ccom 5692  Rel wrel 5693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700

This theorem is referenced by:  f1omvdco2  19480  symgsssg  19499  symgfisg  19500  symggen  19502  pmtrcnel  33091  pmtrcnel2  33092