Theorem f1omvdco3 17090

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 297	. . . . 5 |- ( ( X e. dom ( F \ _I ) <-> X e. dom ( G \ _I ) ) <-> ( -. X e. dom ( F \ _I ) <-> -. X e. dom ( G \ _I ) ) )
2		disjsn 4032	. . . . . . 7 |- ( ( dom ( F \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( F \ _I ) )
3		disj2 3812	. . . . . . 7 |- ( ( dom ( F \ _I ) i^i { X } ) = (/) <-> dom ( F \ _I ) C_ ( _V \ { X } ) )
4	2, 3	bitr3i 255	. . . . . 6 |- ( -. X e. dom ( F \ _I ) <-> dom ( F \ _I ) C_ ( _V \ { X } ) )
5		disjsn 4032	. . . . . . 7 |- ( ( dom ( G \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( G \ _I ) )
6		disj2 3812	. . . . . . 7 |- ( ( dom ( G \ _I ) i^i { X } ) = (/) <-> dom ( G \ _I ) C_ ( _V \ { X } ) )
7	5, 6	bitr3i 255	. . . . . 6 |- ( -. X e. dom ( G \ _I ) <-> dom ( G \ _I ) C_ ( _V \ { X } ) )
8	4, 7	bibi12i 317	. . . . 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 253	. . . 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 298	. . 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 1406	. . 3 |- ( ( X e. dom ( F \ _I ) \/_ X e. dom ( G \ _I ) ) <-> -. ( X e. dom ( F \ _I ) <-> X e. dom ( G \ _I ) ) )
12		df-xor 1406	. . 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 281	. 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 17089	. . 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 3812	. . . . 5 |- ( ( dom ( ( F o. G ) \ _I ) i^i { X } ) = (/) <-> dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) )
16		disjsn 4032	. . . . 5 |- ( ( dom ( ( F o. G ) \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( ( F o. G ) \ _I ) )
17	15, 16	bitr3i 255	. . . 4 |- ( dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) <-> -. X e. dom ( ( F o. G ) \ _I ) )
18	17	con2bii 334	. . 3 |- ( X e. dom ( ( F o. G ) \ _I ) <-> -. dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) )
19	14, 18	sylibr 216	. 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 1306	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 188   /\ w3a 985   \/_ wxo 1405   = wceq 1444   e. wcel 1887   _Vcvv 3045   \ cdif 3401   i^i cin 3403   C_ wss 3404   (/)c0 3731   {csn 3968   _I cid 4744   dom cdm 4834   o. ccom 4838   -1-1-onto->wf1o 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-xor 1406  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590

This theorem is referenced by:  psgnunilem5  17135