Theorem symgfixfo 17859

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 17856	. . 3 |- ( K e. N -> H : Q --> S )
6	5	adantl 482	. 2 |- ( ( N e. V /\ K e. N ) -> H : Q --> S )
7		eqeq1 2626	. . . . . . . . . 10 |- ( i = j -> ( i = K <-> j = K ) )
8		fveq2 6191	. . . . . . . . . 10 |- ( i = j -> ( s ` i ) = ( s ` j ) )
9	7, 8	ifbieq2d 4111	. . . . . . . . 9 |- ( i = j -> if ( i = K , K , ( s ` i ) ) = if ( j = K , K , ( s ` j ) ) )
10	9	cbvmptv 4750	. . . . . . . 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 17858	. . . . . . 7 |- ( ( N e. V /\ K e. N /\ s e. S ) -> ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) e. Q )
12	11	3expa 1265	. . . . . 6 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) e. Q )
13		simpr 477	. . . . . . . . . . . . 13 |- ( ( N e. V /\ K e. N ) -> K e. N )
14	13	anim1i 592	. . . . . . . . . . . 12 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> ( K e. N /\ s e. S ) )
15	14	adantl 482	. . . . . . . . . . 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 2622	. . . . . . . . . . . 12 |- ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) = ( i e. N |-> if ( i = K , K , ( s ` i ) ) )
17	3, 16	symgextres 17845	. . . . . . . . . . 11 |- ( ( K e. N /\ s e. S ) -> ( ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) |` ( N \ { K } ) ) = s )
18	15, 17	syl 17	. . . . . . . . . 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 2628	. . . . . . . . 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 5390	. . . . . . . . . . 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 2632	. . . . . . . . . 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 481	. . . . . . . . 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 247	. . . . . . . 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 450	. . . . . . 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 482	. . . . . 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 3311	. . . . 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 52	. . . 4 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> E. p e. Q s = ( p |` ( N \ { K } ) ) )
28	4	fvtresfn 6284	. . . . . . 7 |- ( p e. Q -> ( H ` p ) = ( p |` ( N \ { K } ) ) )
29	28	eqeq2d 2632	. . . . . 6 |- ( p e. Q -> ( s = ( H ` p ) <-> s = ( p |` ( N \ { K } ) ) ) )
30	29	adantl 482	. . . . 5 |- ( ( ( ( N e. V /\ K e. N ) /\ s e. S ) /\ p e. Q ) -> ( s = ( H ` p ) <-> s = ( p |` ( N \ { K } ) ) ) )
31	30	rexbidva 3049	. . . 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 247	. . 3 |- ( ( ( N e. V /\ K e. N ) /\ s e. S ) -> E. p e. Q s = ( H ` p ) )
33	32	ralrimiva 2966	. 2 |- ( ( N e. V /\ K e. N ) -> A. s e. S E. p e. Q s = ( H ` p ) )
34		dffo3 6374	. 2 |- ( H : Q -onto-> S <-> ( H : Q --> S /\ A. s e. S E. p e. Q s = ( H ` p ) ) )
35	6, 33, 34	sylanbrc 698	1 |- ( ( N e. V /\ K e. N ) -> H : Q -onto-> S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   ifcif 4086   {csn 4177   |-> cmpt 4729   |` cres 5116   -->wf 5884   -onto->wfo 5886   ` cfv 5888   Basecbs 15857   SymGrpcsymg 17797

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  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-int 4476  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-pred 5680  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-tset 15960  df-symg 17798

This theorem is referenced by:  symgfixf1o  17860