Theorem mvdco 19424

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 4454	. . . . . . . 8 ⊢ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I )) = 𝐺
2	1	coeq2i 5840	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = (𝐹 ∘ 𝐺)
3		coundi 6236	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
4	2, 3	eqtr3i 2760	. . . . . 6 ⊢ (𝐹 ∘ 𝐺) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
5	4	difeq1i 4097	. . . . 5 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I )
6		difundir 4266	. . . . 5 ⊢ (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
7	5, 6	eqtri 2758	. . . 4 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
8	7	dmeqi 5884	. . 3 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
9		dmun 5890	. . 3 ⊢ dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
10	8, 9	eqtri 2758	. 2 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
11		inss2 4213	. . . . . 6 ⊢ (𝐺 ∩ I ) ⊆ I
12		coss2 5836	. . . . . 6 ⊢ ((𝐺 ∩ I ) ⊆ I → (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I ))
13	11, 12	ax-mp 5	. . . . 5 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I )
14		cocnvcnv1 6246	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = (𝐹 ∘ I )
15		relcnv 6091	. . . . . . . 8 ⊢ Rel ◡◡𝐹
16		coi1 6251	. . . . . . . 8 ⊢ (Rel ◡◡𝐹 → (◡◡𝐹 ∘ I ) = ◡◡𝐹)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = ◡◡𝐹
18	14, 17	eqtr3i 2760	. . . . . 6 ⊢ (𝐹 ∘ I ) = ◡◡𝐹
19		cnvcnvss 6183	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
20	18, 19	eqsstri 4005	. . . . 5 ⊢ (𝐹 ∘ I ) ⊆ 𝐹
21	13, 20	sstri 3968	. . . 4 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹
22		ssdif 4119	. . . 4 ⊢ ((𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹 → ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ))
23		dmss 5882	. . . 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 4111	. . . . 5 ⊢ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I ))
26		dmss 5882	. . . . 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 5954	. . . 4 ⊢ dom (𝐹 ∘ (𝐺 ∖ I )) ⊆ dom (𝐺 ∖ I )
29	27, 28	sstri 3968	. . 3 ⊢ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )
30		unss12 4163	. . 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 4005	1 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))

Syntax hints:   = wceq 1540   ∖ cdif 3923   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926   I cid 5547  ^◡ccnv 5653  dom cdm 5654   ∘ ccom 5658  Rel wrel 5659

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666

This theorem is referenced by:  f1omvdco2  19427  symgsssg  19446  symgfisg  19447  symggen  19449  pmtrcnel  33046  pmtrcnel2  33047