Theorem mvdco 18576

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 4430	. . . . . . . 8 ⊢ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I )) = 𝐺
2	1	coeq2i 5734	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = (𝐹 ∘ 𝐺)
3		coundi 6103	. . . . . . 7 ⊢ (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
4	2, 3	eqtr3i 2849	. . . . . 6 ⊢ (𝐹 ∘ 𝐺) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
5	4	difeq1i 4098	. . . . 5 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I )
6		difundir 4260	. . . . 5 ⊢ (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
7	5, 6	eqtri 2847	. . . 4 ⊢ ((𝐹 ∘ 𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
8	7	dmeqi 5776	. . 3 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
9		dmun 5782	. . 3 ⊢ dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
10	8, 9	eqtri 2847	. 2 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
11		inss2 4209	. . . . . 6 ⊢ (𝐺 ∩ I ) ⊆ I
12		coss2 5730	. . . . . 6 ⊢ ((𝐺 ∩ I ) ⊆ I → (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I ))
13	11, 12	ax-mp 5	. . . . 5 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I )
14		cocnvcnv1 6113	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = (𝐹 ∘ I )
15		relcnv 5970	. . . . . . . 8 ⊢ Rel ◡◡𝐹
16		coi1 6118	. . . . . . . 8 ⊢ (Rel ◡◡𝐹 → (◡◡𝐹 ∘ I ) = ◡◡𝐹)
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ (◡◡𝐹 ∘ I ) = ◡◡𝐹
18	14, 17	eqtr3i 2849	. . . . . 6 ⊢ (𝐹 ∘ I ) = ◡◡𝐹
19		cnvcnvss 6054	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
20	18, 19	eqsstri 4004	. . . . 5 ⊢ (𝐹 ∘ I ) ⊆ 𝐹
21	13, 20	sstri 3979	. . . 4 ⊢ (𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹
22		ssdif 4119	. . . 4 ⊢ ((𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹 → ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ))
23		dmss 5774	. . . 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 5774	. . . . 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 5845	. . . 4 ⊢ dom (𝐹 ∘ (𝐺 ∖ I )) ⊆ dom (𝐺 ∖ I )
29	27, 28	sstri 3979	. . 3 ⊢ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )
30		unss12 4161	. . 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 4004	1 ⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))

Syntax hints:   = wceq 1536   ∖ cdif 3936   ∪ cun 3937   ∩ cin 3938   ⊆ wss 3939   I cid 5462  ^◡ccnv 5557  dom cdm 5558   ∘ ccom 5562  Rel wrel 5563

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570

This theorem is referenced by:  f1omvdco2  18579  symgsssg  18598  symgfisg  18599  symggen  18601  pmtrcnel  30737  pmtrcnel2  30738