Theorem mvdco 19053

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 4412	. . . . . . . 8 ⊢ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I )) = 𝐺
2	1	coeq2i 5769	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = (𝐹 ∘ 𝐺)
3		coundi 6151	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
4	2, 3	eqtr3i 2768	. . . . . 6 ⊢ (𝐹 ∘ 𝐺) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
5	4	difeq1i 4053	. . . . 5 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I )
6		difundir 4214	. . . . 5 ⊢ (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
7	5, 6	eqtri 2766	. . . 4 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
8	7	dmeqi 5813	. . 3 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
9		dmun 5819	. . 3 ⊢ dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
10	8, 9	eqtri 2766	. 2 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
11		inss2 4163	. . . . . 6 ⊢ (𝐺 ∩ I ) ⊆ I
12		coss2 5765	. . . . . 6 ⊢ ((𝐺 ∩ I ) ⊆ I → (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I ))
13	11, 12	ax-mp 5	. . . . 5 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I )
14		cocnvcnv1 6161	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = (𝐹 ∘ I )
15		relcnv 6012	. . . . . . . 8 ⊢ Rel ◡◡𝐹
16		coi1 6166	. . . . . . . 8 ⊢ (Rel ◡◡𝐹 → (◡◡𝐹 ∘ I ) = ◡◡𝐹)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = ◡◡𝐹
18	14, 17	eqtr3i 2768	. . . . . 6 ⊢ (𝐹 ∘ I ) = ◡◡𝐹
19		cnvcnvss 6097	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
20	18, 19	eqsstri 3955	. . . . 5 ⊢ (𝐹 ∘ I ) ⊆ 𝐹
21	13, 20	sstri 3930	. . . 4 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹
22		ssdif 4074	. . . 4 ⊢ ((𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹 → ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ))
23		dmss 5811	. . . 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 4066	. . . . 5 ⊢ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I ))
26		dmss 5811	. . . . 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 5880	. . . 4 ⊢ dom (𝐹 ∘ (𝐺 ∖ I )) ⊆ dom (𝐺 ∖ I )
29	27, 28	sstri 3930	. . 3 ⊢ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )
30		unss12 4116	. . 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 689	. 2 ⊢ (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
32	10, 31	eqsstri 3955	1 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))

Syntax hints:   = wceq 1539   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887   I cid 5488  ^◡ccnv 5588  dom cdm 5589   ∘ ccom 5593  Rel wrel 5594

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-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601

This theorem is referenced by:  f1omvdco2  19056  symgsssg  19075  symgfisg  19076  symggen  19078  pmtrcnel  31358  pmtrcnel2  31359