Theorem pmtridf1o 29856

Description: Transpositions of X and Y (understood to be the identity when X = Y), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.)

Hypotheses

Ref	Expression
pmtridf1o.a	|- ( ph -> A e. V )
pmtridf1o.x	|- ( ph -> X e. A )
pmtridf1o.y	|- ( ph -> Y e. A )
pmtridf1o.t	|- T = if ( X = Y , ( _I |` A ) , ( (pmTrsp ` A ) ` { X , Y } ) )

Assertion

Ref	Expression
pmtridf1o	|- ( ph -> T : A -1-1-onto-> A )

Proof of Theorem pmtridf1o

Step	Hyp	Ref	Expression
1		pmtridf1o.t	. . . 4 |- T = if ( X = Y , ( _I |` A ) , ( (pmTrsp ` A ) ` { X , Y } ) )
2		iftrue 4092	. . . . 5 |- ( X = Y -> if ( X = Y , ( _I |` A ) , ( (pmTrsp ` A ) ` { X , Y } ) ) = ( _I |` A ) )
3	2	adantl 482	. . . 4 |- ( ( ph /\ X = Y ) -> if ( X = Y , ( _I |` A ) , ( (pmTrsp ` A ) ` { X , Y } ) ) = ( _I |` A ) )
4	1, 3	syl5eq 2668	. . 3 |- ( ( ph /\ X = Y ) -> T = ( _I |` A ) )
5		f1oi 6174	. . . 4 |- ( _I |` A ) : A -1-1-onto-> A
6	5	a1i 11	. . 3 |- ( ( ph /\ X = Y ) -> ( _I |` A ) : A -1-1-onto-> A )
7		f1oeq1 6127	. . . 4 |- ( T = ( _I |` A ) -> ( T : A -1-1-onto-> A <-> ( _I |` A ) : A -1-1-onto-> A ) )
8	7	biimpar 502	. . 3 |- ( ( T = ( _I |` A ) /\ ( _I |` A ) : A -1-1-onto-> A ) -> T : A -1-1-onto-> A )
9	4, 6, 8	syl2anc 693	. 2 |- ( ( ph /\ X = Y ) -> T : A -1-1-onto-> A )
10		simpr 477	. . . . . . 7 |- ( ( ph /\ X =/= Y ) -> X =/= Y )
11	10	neneqd 2799	. . . . . 6 |- ( ( ph /\ X =/= Y ) -> -. X = Y )
12		iffalse 4095	. . . . . 6 |- ( -. X = Y -> if ( X = Y , ( _I |` A ) , ( (pmTrsp ` A ) ` { X , Y } ) ) = ( (pmTrsp ` A ) ` { X , Y } ) )
13	11, 12	syl 17	. . . . 5 |- ( ( ph /\ X =/= Y ) -> if ( X = Y , ( _I |` A ) , ( (pmTrsp ` A ) ` { X , Y } ) ) = ( (pmTrsp ` A ) ` { X , Y } ) )
14	1, 13	syl5eq 2668	. . . 4 |- ( ( ph /\ X =/= Y ) -> T = ( (pmTrsp ` A ) ` { X , Y } ) )
15		pmtridf1o.a	. . . . . 6 |- ( ph -> A e. V )
16	15	adantr 481	. . . . 5 |- ( ( ph /\ X =/= Y ) -> A e. V )
17		pmtridf1o.x	. . . . . . 7 |- ( ph -> X e. A )
18	17	adantr 481	. . . . . 6 |- ( ( ph /\ X =/= Y ) -> X e. A )
19		pmtridf1o.y	. . . . . . 7 |- ( ph -> Y e. A )
20	19	adantr 481	. . . . . 6 |- ( ( ph /\ X =/= Y ) -> Y e. A )
21	18, 20	prssd 4354	. . . . 5 |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ A )
22		pr2nelem 8827	. . . . . 6 |- ( ( X e. A /\ Y e. A /\ X =/= Y ) -> { X , Y } ~~ 2o )
23	18, 20, 10, 22	syl3anc 1326	. . . . 5 |- ( ( ph /\ X =/= Y ) -> { X , Y } ~~ 2o )
24		eqid 2622	. . . . . 6 |- (pmTrsp ` A ) = (pmTrsp ` A )
25		eqid 2622	. . . . . 6 |- ran (pmTrsp ` A ) = ran (pmTrsp ` A )
26	24, 25	pmtrrn 17877	. . . . 5 |- ( ( A e. V /\ { X , Y } C_ A /\ { X , Y } ~~ 2o ) -> ( (pmTrsp ` A ) ` { X , Y } ) e. ran (pmTrsp ` A ) )
27	16, 21, 23, 26	syl3anc 1326	. . . 4 |- ( ( ph /\ X =/= Y ) -> ( (pmTrsp ` A ) ` { X , Y } ) e. ran (pmTrsp ` A ) )
28	14, 27	eqeltrd 2701	. . 3 |- ( ( ph /\ X =/= Y ) -> T e. ran (pmTrsp ` A ) )
29	24, 25	pmtrff1o 17883	. . 3 |- ( T e. ran (pmTrsp ` A ) -> T : A -1-1-onto-> A )
30	28, 29	syl 17	. 2 |- ( ( ph /\ X =/= Y ) -> T : A -1-1-onto-> A )
31	9, 30	pm2.61dane 2881	1 |- ( ph -> T : A -1-1-onto-> A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   ifcif 4086   {cpr 4179   class class class wbr 4653   _I cid 5023   ran crn 5115   |` cres 5116   -1-1-onto->wf1o 5887   ` 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-8 1992  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  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pmtr 17862

This theorem is referenced by:  reprpmtf1o  30704  hgt750lema  30735