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Theorem mvdco 19235
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 4442 . . . . . . . 8 ((𝐺 ∩ I ) ∪ (𝐺 ∖ I )) = 𝐺
21coeq2i 5820 . . . . . . 7 (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = (𝐹𝐺)
3 coundi 6203 . . . . . . 7 (𝐹 ∘ ((𝐺 ∩ I ) ∪ (𝐺 ∖ I ))) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
42, 3eqtr3i 2763 . . . . . 6 (𝐹𝐺) = ((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I )))
54difeq1i 4082 . . . . 5 ((𝐹𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I )
6 difundir 4244 . . . . 5 (((𝐹 ∘ (𝐺 ∩ I )) ∪ (𝐹 ∘ (𝐺 ∖ I ))) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
75, 6eqtri 2761 . . . 4 ((𝐹𝐺) ∖ I ) = (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
87dmeqi 5864 . . 3 dom ((𝐹𝐺) ∖ I ) = dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
9 dmun 5870 . . 3 dom (((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
108, 9eqtri 2761 . 2 dom ((𝐹𝐺) ∖ I ) = (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ))
11 inss2 4193 . . . . . 6 (𝐺 ∩ I ) ⊆ I
12 coss2 5816 . . . . . 6 ((𝐺 ∩ I ) ⊆ I → (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I ))
1311, 12ax-mp 5 . . . . 5 (𝐹 ∘ (𝐺 ∩ I )) ⊆ (𝐹 ∘ I )
14 cocnvcnv1 6213 . . . . . . 7 (𝐹 ∘ I ) = (𝐹 ∘ I )
15 relcnv 6060 . . . . . . . 8 Rel 𝐹
16 coi1 6218 . . . . . . . 8 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1715, 16ax-mp 5 . . . . . . 7 (𝐹 ∘ I ) = 𝐹
1814, 17eqtr3i 2763 . . . . . 6 (𝐹 ∘ I ) = 𝐹
19 cnvcnvss 6150 . . . . . 6 𝐹𝐹
2018, 19eqsstri 3982 . . . . 5 (𝐹 ∘ I ) ⊆ 𝐹
2113, 20sstri 3957 . . . 4 (𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹
22 ssdif 4103 . . . 4 ((𝐹 ∘ (𝐺 ∩ I )) ⊆ 𝐹 → ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ⊆ (𝐹 ∖ I ))
23 dmss 5862 . . . 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 4095 . . . . 5 ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ (𝐹 ∘ (𝐺 ∖ I ))
26 dmss 5862 . . . . 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 5930 . . . 4 dom (𝐹 ∘ (𝐺 ∖ I )) ⊆ dom (𝐺 ∖ I )
2927, 28sstri 3957 . . 3 dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I ) ⊆ dom (𝐺 ∖ I )
30 unss12 4146 . . 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 691 . 2 (dom ((𝐹 ∘ (𝐺 ∩ I )) ∖ I ) ∪ dom ((𝐹 ∘ (𝐺 ∖ I )) ∖ I )) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
3210, 31eqsstri 3982 1 dom ((𝐹𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cdif 3911  cun 3912  cin 3913  wss 3914   I cid 5534  ccnv 5636  dom cdm 5637  ccom 5641  Rel wrel 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649
This theorem is referenced by:  f1omvdco2  19238  symgsssg  19257  symgfisg  19258  symggen  19260  pmtrcnel  31996  pmtrcnel2  31997
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