Theorem symgfixfo 15944

Description: The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is an onto function. (Contributed by AV, 7-Jan-2019.)

Hypotheses

Ref	Expression
symgfixf.p	|- P = ( Base ` ( SymGrp ` N ) )
symgfixf.q	|- Q = { q e. P | ( q ` K ) = K }
symgfixf.s	|- S = ( Base ` ( SymGrp ` ( N \ { K } ) ) )
symgfixf.h	|- H = ( q e. Q |-> ( q |` ( N \ { K } ) ) )

Assertion

Ref	Expression
symgfixfo	|- ( ( N e. V /\ K e. N ) -> H : Q -onto-> S )

Distinct variable groups:   K, q   P, q   N, q   Q, q   S, q

Allowed substitution hints:   H( q)   V( q)

Proof of Theorem symgfixfo

Dummy variables p i s j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		symgfixf.p	. . . 4 |- P = ( Base ` ( SymGrp ` N ) )
2		symgfixf.q	. . . 4 |- Q = { q e. P | ( q ` K ) = K }
3		symgfixf.s	. . . 4 |- S = ( Base ` ( SymGrp ` ( N \ { K } ) ) )
4		symgfixf.h	. . . 4 |- H = ( q e. Q |-> ( q |` ( N \ { K } ) ) )
5	1, 2, 3, 4	symgfixf 15941	. . 3 |- ( K e. N -> H : Q --> S )
6	5	adantl 466	. 2 |- ( ( N e. V /\ K e. N ) -> H : Q --> S )
7		eqeq1 2448	. . . . . . . . . 10 |- ( i = j -> ( i = K <-> j = K ) )
8		fveq2 5690	. . . . . . . . . 10 |- ( i = j -> ( s ` i ) = ( s ` j ) )
9	7, 8	ifbieq2d 3813	. . . . . . . . 9 |- ( i = j -> if ( i = K , K , ( s ` i ) ) = if ( j = K , K , ( s ` j ) ) )
10	9	cbvmptv 4382	. . . . . . . 8 |- ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) = ( j e. N |-> if ( j = K , K , ( s ` j ) ) )
11	1, 2, 3, 4, 10	symgfixfolem1 15943	. . . . . . 7 |- ( ( N e. V /\ K e. N /\ s e. S ) -> ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) e. Q )
12	11	3expa 1187	. . . . . 6 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) e. Q )
13		simpr 461	. . . . . . . . . . . . 13 |- ( ( N e. V /\ K e. N ) -> K e. N )
14	13	anim1i 568	. . . . . . . . . . . 12 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> ( K e. N /\ s e. S ) )
15	14	adantl 466	. . . . . . . . . . 11 |- ( ( p = ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) /\ ( ( N e. V /\ K e. N ) /\ s e. S ) ) -> ( K e. N /\ s e. S ) )
16		eqid 2442	. . . . . . . . . . . 12 |- ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) = ( i e. N |-> if ( i = K , K , ( s ` i ) ) )
17	3, 16	symgextres 15929	. . . . . . . . . . 11 |- ( ( K e. N /\ s e. S ) -> ( ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) |` ( N \ { K } ) ) = s )
18	15, 17	syl 16	. . . . . . . . . 10 |- ( ( p = ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) /\ ( ( N e. V /\ K e. N ) /\ s e. S ) ) -> ( ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) |` ( N \ { K } ) ) = s )
19	18	eqcomd 2447	. . . . . . . . 9 |- ( ( p = ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) /\ ( ( N e. V /\ K e. N ) /\ s e. S ) ) -> s = ( ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) |` ( N \ { K } ) ) )
20		reseq1 5103	. . . . . . . . . . 11 |- ( p = ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) -> ( p |` ( N \ { K } ) ) = ( ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) |` ( N \ { K } ) ) )
21	20	eqeq2d 2453	. . . . . . . . . 10 |- ( p = ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) -> ( s = ( p |` ( N \ { K } ) ) <-> s = ( ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) |` ( N \ { K } ) ) ) )
22	21	adantr 465	. . . . . . . . 9 |- ( ( p = ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) /\ ( ( N e. V /\ K e. N ) /\ s e. S ) ) -> ( s = ( p |` ( N \ { K } ) ) <-> s = ( ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) |` ( N \ { K } ) ) ) )
23	19, 22	mpbird 232	. . . . . . . 8 |- ( ( p = ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) /\ ( ( N e. V /\ K e. N ) /\ s e. S ) ) -> s = ( p |` ( N \ { K } ) ) )
24	23	ex 434	. . . . . . 7 |- ( p = ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) -> ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> s = ( p |` ( N \ { K } ) ) ) )
25	24	adantl 466	. . . . . 6 |- ( ( ( ( N e. V /\ K e. N ) /\ s e. S ) /\ p = ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) ) -> ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> s = ( p |` ( N \ { K } ) ) ) )
26	12, 25	rspcimedv 3074	. . . . 5 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> E. p e. Q s = ( p |` ( N \ { K } ) ) ) )
27	26	pm2.43i 47	. . . 4 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> E. p e. Q s = ( p |` ( N \ { K } ) ) )
28	4	fvtresfn 5774	. . . . . . 7 |- ( p e. Q -> ( H ` p ) = ( p |` ( N \ { K } ) ) )
29	28	eqeq2d 2453	. . . . . 6 |- ( p e. Q -> ( s = ( H ` p ) <-> s = ( p |` ( N \ { K } ) ) ) )
30	29	adantl 466	. . . . 5 |- ( ( ( ( N e. V /\ K e. N ) /\ s e. S ) /\ p e. Q ) -> ( s = ( H ` p ) <-> s = ( p |` ( N \ { K } ) ) ) )
31	30	rexbidva 2731	. . . 4 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> ( E. p e. Q s = ( H ` p ) <-> E. p e. Q s = ( p |` ( N \ { K } ) ) ) )
32	27, 31	mpbird 232	. . 3 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> E. p e. Q s = ( H ` p ) )
33	32	ralrimiva 2798	. 2 |- ( ( N e. V /\ K e. N ) -> A. s e. S E. p e. Q s = ( H ` p ) )
34		dffo3 5857	. 2 |- ( H : Q -onto-> S <-> ( H : Q --> S /\ A. s e. S E. p e. Q s = ( H ` p ) ) )
35	6, 33, 34	sylanbrc 664	1 |- ( ( N e. V /\ K e. N ) -> H : Q -onto-> S )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2714   E.wrex 2715   {crab 2718   \ cdif 3324   ifcif 3790   {csn 3876   e. cmpt 4349   |` cres 4841   -->wf 5413   -onto->wfo 5415   ` cfv 5417   Basecbs 14173   SymGrpcsymg 15881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-plusg 14250  df-tset 14256  df-symg 15882

This theorem is referenced by:  symgfixf1o  15945