Theorem f1omvdco3 16347

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 295	. . . . 5 |- ( ( X e. dom ( F \ _I ) <-> X e. dom ( G \ _I ) ) <-> ( -. X e. dom ( F \ _I ) <-> -. X e. dom ( G \ _I ) ) )
2		disjsn 4094	. . . . . . 7 |- ( ( dom ( F \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( F \ _I ) )
3		disj2 3879	. . . . . . 7 |- ( ( dom ( F \ _I ) i^i { X } ) = (/) <-> dom ( F \ _I ) C_ ( _V \ { X } ) )
4	2, 3	bitr3i 251	. . . . . 6 |- ( -. X e. dom ( F \ _I ) <-> dom ( F \ _I ) C_ ( _V \ { X } ) )
5		disjsn 4094	. . . . . . 7 |- ( ( dom ( G \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( G \ _I ) )
6		disj2 3879	. . . . . . 7 |- ( ( dom ( G \ _I ) i^i { X } ) = (/) <-> dom ( G \ _I ) C_ ( _V \ { X } ) )
7	5, 6	bitr3i 251	. . . . . 6 |- ( -. X e. dom ( G \ _I ) <-> dom ( G \ _I ) C_ ( _V \ { X } ) )
8	4, 7	bibi12i 315	. . . . 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 249	. . . 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 296	. . 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 1361	. . 3 |- ( ( X e. dom ( F \ _I ) \/_ X e. dom ( G \ _I ) ) <-> -. ( X e. dom ( F \ _I ) <-> X e. dom ( G \ _I ) ) )
12		df-xor 1361	. . 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 277	. 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 16346	. . 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 3879	. . . . 5 |- ( ( dom ( ( F o. G ) \ _I ) i^i { X } ) = (/) <-> dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) )
16		disjsn 4094	. . . . 5 |- ( ( dom ( ( F o. G ) \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( ( F o. G ) \ _I ) )
17	15, 16	bitr3i 251	. . . 4 |- ( dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) <-> -. X e. dom ( ( F o. G ) \ _I ) )
18	17	con2bii 332	. . 3 |- ( X e. dom ( ( F o. G ) \ _I ) <-> -. dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) )
19	14, 18	sylibr 212	. 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 1266	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 184   /\ w3a 973   \/_ wxo 1360   = wceq 1379   e. wcel 1767   _Vcvv 3118   \ cdif 3478   i^i cin 3480   C_ wss 3481   (/)c0 3790   {csn 4033   _I cid 4796   dom cdm 5005   o. ccom 5009   -1-1-onto->wf1o 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602

This theorem is referenced by:  psgnunilem5  16392