Theorem pmtrrn 17877

Description: Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Hypotheses

Ref	Expression
pmtrrn.t	|- T = (pmTrsp ` D )
pmtrrn.r	|- R = ran T

Assertion

Ref	Expression
pmtrrn	|- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( T ` P ) e. R )

Proof of Theorem pmtrrn

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptexg 6484	. . . . . . 7 |- ( D e. V -> ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) e. _V )
2	1	ralrimivw 2967	. . . . . 6 |- ( D e. V -> A. z e. { x e. ~P D | x ~~ 2o } ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) e. _V )
3	2	3ad2ant1 1082	. . . . 5 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> A. z e. { x e. ~P D | x ~~ 2o } ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) e. _V )
4		eqid 2622	. . . . . 6 |- ( z e. { x e. ~P D | x ~~ 2o } |-> ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) ) = ( z e. { x e. ~P D | x ~~ 2o } |-> ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) )
5	4	fnmpt 6020	. . . . 5 |- ( A. z e. { x e. ~P D | x ~~ 2o } ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) e. _V -> ( z e. { x e. ~P D | x ~~ 2o } |-> ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) ) Fn { x e. ~P D | x ~~ 2o } )
6	3, 5	syl 17	. . . 4 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( z e. { x e. ~P D | x ~~ 2o } |-> ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) ) Fn { x e. ~P D | x ~~ 2o } )
7		pmtrrn.t	. . . . . . 7 |- T = (pmTrsp ` D )
8	7	pmtrfval 17870	. . . . . 6 |- ( D e. V -> T = ( z e. { x e. ~P D | x ~~ 2o } |-> ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) ) )
9	8	3ad2ant1 1082	. . . . 5 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> T = ( z e. { x e. ~P D | x ~~ 2o } |-> ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) ) )
10	9	fneq1d 5981	. . . 4 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( T Fn { x e. ~P D | x ~~ 2o } <-> ( z e. { x e. ~P D | x ~~ 2o } |-> ( y e. D |-> if ( y e. z , U. ( z \ { y } ) , y ) ) ) Fn { x e. ~P D | x ~~ 2o } ) )
11	6, 10	mpbird 247	. . 3 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> T Fn { x e. ~P D | x ~~ 2o } )
12		elpw2g 4827	. . . . . 6 |- ( D e. V -> ( P e. ~P D <-> P C_ D ) )
13	12	biimpar 502	. . . . 5 |- ( ( D e. V /\ P C_ D ) -> P e. ~P D )
14	13	3adant3 1081	. . . 4 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> P e. ~P D )
15		simp3 1063	. . . 4 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> P ~~ 2o )
16		breq1 4656	. . . . 5 |- ( x = P -> ( x ~~ 2o <-> P ~~ 2o ) )
17	16	elrab 3363	. . . 4 |- ( P e. { x e. ~P D | x ~~ 2o } <-> ( P e. ~P D /\ P ~~ 2o ) )
18	14, 15, 17	sylanbrc 698	. . 3 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> P e. { x e. ~P D | x ~~ 2o } )
19		fnfvelrn 6356	. . 3 |- ( ( T Fn { x e. ~P D | x ~~ 2o } /\ P e. { x e. ~P D | x ~~ 2o } ) -> ( T ` P ) e. ran T )
20	11, 18, 19	syl2anc 693	. 2 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( T ` P ) e. ran T )
21		pmtrrn.r	. 2 |- R = ran T
22	20, 21	syl6eleqr 2712	1 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( T ` P ) e. R )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   ` cfv 5888   2oc2o 7554   ~~ cen 7952  pmTrspcpmtr 17861

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  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-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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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  df-pmtr 17862

This theorem is referenced by:  pmtrfb  17885  symggen  17890  pmtr3ncom  17895  pmtrdifellem1  17896  mdetralt  20414  pmtrto1cl  29849  pmtridf1o  29856