Theorem symgfixfo 16255

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 16252	. . 3 |- ( K e. N -> H : Q --> S )
6	5	adantl 466	. 2 |- ( ( N e. V /\ K e. N ) -> H : Q --> S )
7		eqeq1 2466	. . . . . . . . . 10 |- ( i = j -> ( i = K <-> j = K ) )
8		fveq2 5859	. . . . . . . . . 10 |- ( i = j -> ( s ` i ) = ( s ` j ) )
9	7, 8	ifbieq2d 3959	. . . . . . . . 9 |- ( i = j -> if ( i = K , K , ( s ` i ) ) = if ( j = K , K , ( s ` j ) ) )
10	9	cbvmptv 4533	. . . . . . . 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 16254	. . . . . . 7 |- ( ( N e. V /\ K e. N /\ s e. S ) -> ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) e. Q )
12	11	3expa 1191	. . . . . 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 2462	. . . . . . . . . . . 12 |- ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) = ( i e. N |-> if ( i = K , K , ( s ` i ) ) )
17	3, 16	symgextres 16240	. . . . . . . . . . 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 2470	. . . . . . . . 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 5260	. . . . . . . . . . 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 2476	. . . . . . . . . 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 3211	. . . . 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 5944	. . . . . . 7 |- ( p e. Q -> ( H ` p ) = ( p |` ( N \ { K } ) ) )
29	28	eqeq2d 2476	. . . . . 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 2965	. . . 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 2873	. 2 |- ( ( N e. V /\ K e. N ) -> A. s e. S E. p e. Q s = ( H ` p ) )
34		dffo3 6029	. 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 1374   e. wcel 1762   A.wral 2809   E.wrex 2810   {crab 2813   \ cdif 3468   ifcif 3934   {csn 4022   |-> cmpt 4500   |` cres 4996   -->wf 5577   -onto->wfo 5579   ` cfv 5581   Basecbs 14481   SymGrpcsymg 16192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-plusg 14559  df-tset 14565  df-symg 16193

This theorem is referenced by:  symgfixf1o  16256