Theorem f1omvdco3 16077

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 4047	. . . . . . 7 |- ( ( dom ( F \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( F \ _I ) )
3		disj2 3837	. . . . . . 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 4047	. . . . . . 7 |- ( ( dom ( G \ _I ) i^i { X } ) = (/) <-> -. X e. dom ( G \ _I ) )
6		disj2 3837	. . . . . . 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 1352	. . 3 |- ( ( X e. dom ( F \ _I ) \/_ X e. dom ( G \ _I ) ) <-> -. ( X e. dom ( F \ _I ) <-> X e. dom ( G \ _I ) ) )
12		df-xor 1352	. . 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 16076	. . 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 3837	. . . . 5 |- ( ( dom ( ( F o. G ) \ _I ) i^i { X } ) = (/) <-> dom ( ( F o. G ) \ _I ) C_ ( _V \ { X } ) )
16		disjsn 4047	. . . . 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 1257	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 965   \/_ wxo 1351   = wceq 1370   e. wcel 1758   _Vcvv 3078   \ cdif 3436   i^i cin 3438   C_ wss 3439   (/)c0 3748   {csn 3988   _I cid 4742   dom cdm 4951   o. ccom 4955   -1-1-onto->wf1o 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-xor 1352  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537

This theorem is referenced by:  psgnunilem5  16122