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Theorem mvdco 18130
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 4206 . . . . . . . 8 ((𝐺 ∩ I ) ∪ (𝐺 ∖ I )) = 𝐺
21coeq2i 5451 . . . . . . 7 (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = (𝐹𝐺)
3 coundi 5822 . . . . . . 7 (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
42, 3eqtr3i 2789 . . . . . 6 (𝐹𝐺) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
54difeq1i 3886 . . . . 5 ((𝐹𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I )
6 difundir 4045 . . . . 5 (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
75, 6eqtri 2787 . . . 4 ((𝐹𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
87dmeqi 5493 . . 3 dom ((𝐹𝐺) ∖ I ) = dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
9 dmun 5499 . . 3 dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
108, 9eqtri 2787 . 2 dom ((𝐹𝐺) ∖ I ) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
11 inss2 3993 . . . . . 6 (𝐺 ∩ I ) ⊆ I
12 coss2 5447 . . . . . 6 ((𝐺 ∩ I ) ⊆ I → (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I ))
1311, 12ax-mp 5 . . . . 5 (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I )
14 cocnvcnv1 5832 . . . . . . 7 (𝐹 ∘ I ) = (𝐹 ∘ I )
15 relcnv 5685 . . . . . . . 8 Rel 𝐹
16 coi1 5837 . . . . . . . 8 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1715, 16ax-mp 5 . . . . . . 7 (𝐹 ∘ I ) = 𝐹
1814, 17eqtr3i 2789 . . . . . 6 (𝐹 ∘ I ) = 𝐹
19 cnvcnvss 5772 . . . . . 6 𝐹𝐹
2018, 19eqsstri 3795 . . . . 5 (𝐹 ∘ I ) ⊆ 𝐹
2113, 20sstri 3770 . . . 4 (𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹
22 ssdif 3907 . . . 4 ((𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹 → ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ))
23 dmss 5491 . . . 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 3899 . . . . 5 ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I ))
26 dmss 5491 . . . . 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 5554 . . . 4 dom (𝐹 ∘ (𝐺 ∖ I )) ⊆ dom (𝐺 ∖ I )
2927, 28sstri 3770 . . 3 dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )
30 unss12 3947 . . 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 683 . 2 (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
3210, 31eqsstri 3795 1 dom ((𝐹𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cdif 3729  cun 3730  cin 3731  wss 3732   I cid 5184  ccnv 5276  dom cdm 5277  ccom 5281  Rel wrel 5282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289
This theorem is referenced by:  f1omvdco2  18133  symgsssg  18152  symgfisg  18153  symggen  18155
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