Theorem f1omvdco3 17869

Description: If a point is moved by exactly one of two permutations, then it will be moved by their composite. (Contributed by Stefan O'Rear, 23-Aug-2015.)

Assertion

Ref	Expression
f1omvdco3	|- ( ( F : A -1-1-onto-> A /\ G : A -1-1-onto-> A /\ ( X e. dom ( F \ _I ) \/_ X e. dom ( G \ _I ) ) ) -> X e. dom ( ( F o. G ) \ _I ) )

Proof of Theorem f1omvdco3

Step	Hyp	Ref	Expression
1		notbi 309	. . . . 5 |- ( ( X e. dom ( F \ _I ) <-> X e. dom ( G \ _I ) ) <-> ( -. X e. dom ( F \ _I ) <-> -. X e. dom ( G \ _I ) ) )
2		disjsn 4246	. . . . . . 7 |- ( ( dom ( F \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( F \ _I ) )
3		disj2 4024	. . . . . . 7 |- ( ( dom ( F \ _I ) i^i { X } ) = (/) <-> dom ( F \ _I ) C_ ( _V \ { X } ) )
4	2, 3	bitr3i 266	. . . . . 6 |- ( -. X e. dom ( F \ _I ) <-> dom ( F \ _I ) C_ ( _V \ { X } ) )
5		disjsn 4246	. . . . . . 7 |- ( ( dom ( G \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( G \ _I ) )
6		disj2 4024	. . . . . . 7 |- ( ( dom ( G \ _I ) i^i { X } ) = (/) <-> dom ( G \ _I ) C_ ( _V \ { X } ) )
7	5, 6	bitr3i 266	. . . . . 6 |- ( -. X e. dom ( G \ _I ) <-> dom ( G \ _I ) C_ ( _V \ { X } ) )
8	4, 7	bibi12i 329	. . . . 5 |- ( ( -. X e. dom ( F \ _I ) <-> -. X e. dom ( G \ _I ) ) <-> ( dom ( F \ _I ) C_ ( _V \ { X } ) <-> dom ( G \ _I ) C_ ( _V \ { X } ) ) )
9	1, 8	bitri 264	. . . 4 |- ( ( X e. dom ( F \ _I ) <-> X e. dom ( G \ _I ) ) <-> ( dom ( F \ _I ) C_ ( _V \ { X } ) <-> dom ( G \ _I ) C_ ( _V \ { X } ) ) )
10	9	notbii 310	. . 3 |- ( -. ( X e. dom ( F \ _I ) <-> X e. dom ( G \ _I ) ) <-> -. ( dom ( F \ _I ) C_ ( _V \ { X } ) <-> dom ( G \ _I ) C_ ( _V \ { X } ) ) )
11		df-xor 1465	. . 3 |- ( ( X e. dom ( F \ _I ) \/_ X e. dom ( G \ _I ) ) <-> -. ( X e. dom ( F \ _I ) <-> X e. dom ( G \ _I ) ) )
12		df-xor 1465	. . 3 |- ( ( dom ( F \ _I ) C_ ( _V \ { X } ) \/_ dom ( G \ _I ) C_ ( _V \ { X } ) ) <-> -. ( dom ( F \ _I ) C_ ( _V \ { X } ) <-> dom ( G \ _I ) C_ ( _V \ { X } ) ) )
13	10, 11, 12	3bitr4i 292	. 2 |- ( ( X e. dom ( F \ _I ) \/_ X e. dom ( G \ _I ) ) <-> ( dom ( F \ _I ) C_ ( _V \ { X } ) \/_ dom ( G \ _I ) C_ ( _V \ { X } ) ) )
14		f1omvdco2 17868	. . 3 |- ( ( F : A -1-1-onto-> A /\ G : A -1-1-onto-> A /\ ( dom ( F \ _I ) C_ ( _V \ { X } ) \/_ dom ( G \ _I ) C_ ( _V \ { X } ) ) ) -> -. dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) )
15		disj2 4024	. . . . 5 |- ( ( dom ( ( F o. G ) \ _I ) i^i { X } ) = (/) <-> dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) )
16		disjsn 4246	. . . . 5 |- ( ( dom ( ( F o. G ) \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( ( F o. G ) \ _I ) )
17	15, 16	bitr3i 266	. . . 4 |- ( dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) <-> -. X e. dom ( ( F o. G ) \ _I ) )
18	17	con2bii 347	. . 3 |- ( X e. dom ( ( F o. G ) \ _I ) <-> -. dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) )
19	14, 18	sylibr 224	. 2 |- ( ( F : A -1-1-onto-> A /\ G : A -1-1-onto-> A /\ ( dom ( F \ _I ) C_ ( _V \ { X } ) \/_ dom ( G \ _I ) C_ ( _V \ { X } ) ) ) -> X e. dom ( ( F o. G ) \ _I ) )
20	13, 19	syl3an3b 1364	1 |- ( ( F : A -1-1-onto-> A /\ G : A -1-1-onto-> A /\ ( X e. dom ( F \ _I ) \/_ X e. dom ( G \ _I ) ) ) -> X e. dom ( ( F o. G ) \ _I ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ w3a 1037   \/_ wxo 1464   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   _I cid 5023   dom cdm 5114   o. ccom 5118   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-xor 1465  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  psgnunilem5  17914