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Theorem mvdco 18493
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 4430 . . . . . . . 8 ((𝐺 ∩ I ) ∪ (𝐺 ∖ I )) = 𝐺
21coeq2i 5730 . . . . . . 7 (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = (𝐹𝐺)
3 coundi 6098 . . . . . . 7 (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
42, 3eqtr3i 2851 . . . . . 6 (𝐹𝐺) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
54difeq1i 4099 . . . . 5 ((𝐹𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I )
6 difundir 4261 . . . . 5 (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
75, 6eqtri 2849 . . . 4 ((𝐹𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
87dmeqi 5772 . . 3 dom ((𝐹𝐺) ∖ I ) = dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
9 dmun 5778 . . 3 dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
108, 9eqtri 2849 . 2 dom ((𝐹𝐺) ∖ I ) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
11 inss2 4210 . . . . . 6 (𝐺 ∩ I ) ⊆ I
12 coss2 5726 . . . . . 6 ((𝐺 ∩ I ) ⊆ I → (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I ))
1311, 12ax-mp 5 . . . . 5 (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I )
14 cocnvcnv1 6108 . . . . . . 7 (𝐹 ∘ I ) = (𝐹 ∘ I )
15 relcnv 5965 . . . . . . . 8 Rel 𝐹
16 coi1 6113 . . . . . . . 8 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1715, 16ax-mp 5 . . . . . . 7 (𝐹 ∘ I ) = 𝐹
1814, 17eqtr3i 2851 . . . . . 6 (𝐹 ∘ I ) = 𝐹
19 cnvcnvss 6049 . . . . . 6 𝐹𝐹
2018, 19eqsstri 4005 . . . . 5 (𝐹 ∘ I ) ⊆ 𝐹
2113, 20sstri 3980 . . . 4 (𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹
22 ssdif 4120 . . . 4 ((𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹 → ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ))
23 dmss 5770 . . . 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 4112 . . . . 5 ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I ))
26 dmss 5770 . . . . 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 5841 . . . 4 dom (𝐹 ∘ (𝐺 ∖ I )) ⊆ dom (𝐺 ∖ I )
2927, 28sstri 3980 . . 3 dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )
30 unss12 4162 . . 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 688 . 2 (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
3210, 31eqsstri 4005 1 dom ((𝐹𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  cdif 3937  cun 3938  cin 3939  wss 3940   I cid 5458  ccnv 5553  dom cdm 5554  ccom 5558  Rel wrel 5559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566
This theorem is referenced by:  f1omvdco2  18496  symgsssg  18515  symgfisg  18516  symggen  18518  pmtrcnel  30650  pmtrcnel2  30651
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