Theorem symgfixfo 16591

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 16588	. . 3 |- ( K e. N -> H : Q --> S )
6	5	adantl 466	. 2 |- ( ( N e. V /\ K e. N ) -> H : Q --> S )
7		eqeq1 2461	. . . . . . . . . 10 |- ( i = j -> ( i = K <-> j = K ) )
8		fveq2 5872	. . . . . . . . . 10 |- ( i = j -> ( s ` i ) = ( s ` j ) )
9	7, 8	ifbieq2d 3969	. . . . . . . . 9 |- ( i = j -> if ( i = K , K , ( s ` i ) ) = if ( j = K , K , ( s ` j ) ) )
10	9	cbvmptv 4548	. . . . . . . 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 16590	. . . . . . 7 |- ( ( N e. V /\ K e. N /\ s e. S ) -> ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) e. Q )
12	11	3expa 1196	. . . . . 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 2457	. . . . . . . . . . . 12 |- ( i e. N |-> if ( i = K , K , ( s ` i ) ) ) = ( i e. N |-> if ( i = K , K , ( s ` i ) ) )
17	3, 16	symgextres 16577	. . . . . . . . . . 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 2465	. . . . . . . . 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 5277	. . . . . . . . . . 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 2471	. . . . . . . . . 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 3212	. . . . 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 5957	. . . . . . 7 |- ( p e. Q -> ( H ` p ) = ( p |` ( N \ { K } ) ) )
29	28	eqeq2d 2471	. . . . . 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 2871	. 2 |- ( ( N e. V /\ K e. N ) -> A. s e. S E. p e. Q s = ( H ` p ) )
34		dffo3 6047	. 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 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   \ cdif 3468   ifcif 3944   {csn 4032   |-> cmpt 4515   |` cres 5010   -->wf 5590   -onto->wfo 5592   ` cfv 5594   Basecbs 14644   SymGrpcsymg 16529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-plusg 14725  df-tset 14731  df-symg 16530

This theorem is referenced by:  symgfixf1o  16592