Theorem mvdco 19312

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 4478	. . . . . . . 8 ⊢ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I )) = 𝐺
2	1	coeq2i 5860	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = (𝐹 ∘ 𝐺)
3		coundi 6246	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
4	2, 3	eqtr3i 2762	. . . . . 6 ⊢ (𝐹 ∘ 𝐺) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
5	4	difeq1i 4118	. . . . 5 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I )
6		difundir 4280	. . . . 5 ⊢ (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
7	5, 6	eqtri 2760	. . . 4 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
8	7	dmeqi 5904	. . 3 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
9		dmun 5910	. . 3 ⊢ dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
10	8, 9	eqtri 2760	. 2 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
11		inss2 4229	. . . . . 6 ⊢ (𝐺 ∩ I ) ⊆ I
12		coss2 5856	. . . . . 6 ⊢ ((𝐺 ∩ I ) ⊆ I → (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I ))
13	11, 12	ax-mp 5	. . . . 5 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I )
14		cocnvcnv1 6256	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = (𝐹 ∘ I )
15		relcnv 6103	. . . . . . . 8 ⊢ Rel ◡◡𝐹
16		coi1 6261	. . . . . . . 8 ⊢ (Rel ◡◡𝐹 → (◡◡𝐹 ∘ I ) = ◡◡𝐹)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = ◡◡𝐹
18	14, 17	eqtr3i 2762	. . . . . 6 ⊢ (𝐹 ∘ I ) = ◡◡𝐹
19		cnvcnvss 6193	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
20	18, 19	eqsstri 4016	. . . . 5 ⊢ (𝐹 ∘ I ) ⊆ 𝐹
21	13, 20	sstri 3991	. . . 4 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹
22		ssdif 4139	. . . 4 ⊢ ((𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹 → ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ))
23		dmss 5902	. . . 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 4131	. . . . 5 ⊢ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I ))
26		dmss 5902	. . . . 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 5970	. . . 4 ⊢ dom (𝐹 ∘ (𝐺 ∖ I )) ⊆ dom (𝐺 ∖ I )
29	27, 28	sstri 3991	. . 3 ⊢ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )
30		unss12 4182	. . 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 690	. 2 ⊢ (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
32	10, 31	eqsstri 4016	1 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))

Syntax hints:   = wceq 1541   ∖ cdif 3945   ∪ cun 3946   ∩ cin 3947   ⊆ wss 3948   I cid 5573  ^◡ccnv 5675  dom cdm 5676   ∘ ccom 5680  Rel wrel 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688

This theorem is referenced by:  f1omvdco2  19315  symgsssg  19334  symgfisg  19335  symggen  19337  pmtrcnel  32245  pmtrcnel2  32246