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Theorem mvdco 18073
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 ((𝐹𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))

Proof of Theorem mvdco
StepHypRef Expression
1 inundif 4189 . . . . . . . 8 ((𝐺 ∩ I ) ∪ (𝐺 ∖ I )) = 𝐺
21coeq2i 5422 . . . . . . 7 (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = (𝐹𝐺)
3 coundi 5781 . . . . . . 7 (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
42, 3eqtr3i 2795 . . . . . 6 (𝐹𝐺) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
54difeq1i 3876 . . . . 5 ((𝐹𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I )
6 difundir 4030 . . . . 5 (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
75, 6eqtri 2793 . . . 4 ((𝐹𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
87dmeqi 5464 . . 3 dom ((𝐹𝐺) ∖ I ) = dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
9 dmun 5470 . . 3 dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
108, 9eqtri 2793 . 2 dom ((𝐹𝐺) ∖ I ) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
11 inss2 3983 . . . . . 6 (𝐺 ∩ I ) ⊆ I
12 coss2 5418 . . . . . 6 ((𝐺 ∩ I ) ⊆ I → (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I ))
1311, 12ax-mp 5 . . . . 5 (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I )
14 cocnvcnv1 5791 . . . . . . 7 (𝐹 ∘ I ) = (𝐹 ∘ I )
15 relcnv 5645 . . . . . . . 8 Rel 𝐹
16 coi1 5796 . . . . . . . 8 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1715, 16ax-mp 5 . . . . . . 7 (𝐹 ∘ I ) = 𝐹
1814, 17eqtr3i 2795 . . . . . 6 (𝐹 ∘ I ) = 𝐹
19 cnvcnvss 5731 . . . . . 6 𝐹𝐹
2018, 19eqsstri 3785 . . . . 5 (𝐹 ∘ I ) ⊆ 𝐹
2113, 20sstri 3762 . . . 4 (𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹
22 ssdif 3897 . . . 4 ((𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹 → ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ))
23 dmss 5462 . . . 4 (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ) → dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ dom (𝐹 ∖ I ))
2421, 22, 23mp2b 10 . . 3 dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ dom (𝐹 ∖ I )
25 difss 3889 . . . . 5 ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I ))
26 dmss 5462 . . . . 5 (((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I )) → dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐹 ∘ (𝐺 ∖ I )))
2725, 26ax-mp 5 . . . 4 dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐹 ∘ (𝐺 ∖ I ))
28 dmcoss 5524 . . . 4 dom (𝐹 ∘ (𝐺 ∖ I )) ⊆ dom (𝐺 ∖ I )
2927, 28sstri 3762 . . 3 dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )
30 unss12 3937 . . 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 )))
3124, 29, 30mp2an 666 . 2 (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
3210, 31eqsstri 3785 1 dom ((𝐹𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cdif 3721  cun 3722  cin 3723  wss 3724   I cid 5157  ccnv 5249  dom cdm 5250  ccom 5254  Rel wrel 5255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262
This theorem is referenced by:  f1omvdco2  18076  symgsssg  18095  symgfisg  18096  symggen  18098
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