Theorem symgfix2 17836

Description: If a permutation does not move a certain element of a set to a second element, there is a third element which is moved to the second element. (Contributed by AV, 2-Jan-2019.)

Hypothesis

Ref	Expression
symgfix2.p	|- P = ( Base ` ( SymGrp ` N ) )

Assertion

Ref	Expression
symgfix2	|- ( L e. N -> ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) )

Distinct variable groups:   k, N   Q, k   k, K, q   k, L, q   P, q   Q, q

Allowed substitution hints:   P( k)   N( q)

Proof of Theorem symgfix2

Dummy variable l is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3584	. . 3 |- ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) <-> ( Q e. P /\ -. Q e. { q e. P | ( q ` K ) = L } ) )
2		ianor 509	. . . . 5 |- ( -. ( Q e. P /\ ( Q ` K ) = L ) <-> ( -. Q e. P \/ -. ( Q ` K ) = L ) )
3		fveq1 6190	. . . . . . 7 |- ( q = Q -> ( q ` K ) = ( Q ` K ) )
4	3	eqeq1d 2624	. . . . . 6 |- ( q = Q -> ( ( q ` K ) = L <-> ( Q ` K ) = L ) )
5	4	elrab 3363	. . . . 5 |- ( Q e. { q e. P | ( q ` K ) = L } <-> ( Q e. P /\ ( Q ` K ) = L ) )
6	2, 5	xchnxbir 323	. . . 4 |- ( -. Q e. { q e. P | ( q ` K ) = L } <-> ( -. Q e. P \/ -. ( Q ` K ) = L ) )
7	6	anbi2i 730	. . 3 |- ( ( Q e. P /\ -. Q e. { q e. P | ( q ` K ) = L } ) <-> ( Q e. P /\ ( -. Q e. P \/ -. ( Q ` K ) = L ) ) )
8	1, 7	bitri 264	. 2 |- ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) <-> ( Q e. P /\ ( -. Q e. P \/ -. ( Q ` K ) = L ) ) )
9		pm2.21 120	. . . . 5 |- ( -. Q e. P -> ( Q e. P -> ( L e. N -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) ) )
10		symgfix2.p	. . . . . . 7 |- P = ( Base ` ( SymGrp ` N ) )
11	10	symgmov2 17813	. . . . . 6 |- ( Q e. P -> A. l e. N E. k e. N ( Q ` k ) = l )
12		eqeq2 2633	. . . . . . . . . . 11 |- ( l = L -> ( ( Q ` k ) = l <-> ( Q ` k ) = L ) )
13	12	rexbidv 3052	. . . . . . . . . 10 |- ( l = L -> ( E. k e. N ( Q ` k ) = l <-> E. k e. N ( Q ` k ) = L ) )
14	13	rspcva 3307	. . . . . . . . 9 |- ( ( L e. N /\ A. l e. N E. k e. N ( Q ` k ) = l ) -> E. k e. N ( Q ` k ) = L )
15		eqeq2 2633	. . . . . . . . . . . . . . . 16 |- ( L = ( Q ` k ) -> ( ( Q ` K ) = L <-> ( Q ` K ) = ( Q ` k ) ) )
16	15	eqcoms 2630	. . . . . . . . . . . . . . 15 |- ( ( Q ` k ) = L -> ( ( Q ` K ) = L <-> ( Q ` K ) = ( Q ` k ) ) )
17	16	notbid 308	. . . . . . . . . . . . . 14 |- ( ( Q ` k ) = L -> ( -. ( Q ` K ) = L <-> -. ( Q ` K ) = ( Q ` k ) ) )
18		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( K = k -> ( Q ` K ) = ( Q ` k ) )
19	18	eqcoms 2630	. . . . . . . . . . . . . . 15 |- ( k = K -> ( Q ` K ) = ( Q ` k ) )
20	19	necon3bi 2820	. . . . . . . . . . . . . 14 |- ( -. ( Q ` K ) = ( Q ` k ) -> k =/= K )
21	17, 20	syl6bi 243	. . . . . . . . . . . . 13 |- ( ( Q ` k ) = L -> ( -. ( Q ` K ) = L -> k =/= K ) )
22	21	com12 32	. . . . . . . . . . . 12 |- ( -. ( Q ` K ) = L -> ( ( Q ` k ) = L -> k =/= K ) )
23	22	pm4.71rd 667	. . . . . . . . . . 11 |- ( -. ( Q ` K ) = L -> ( ( Q ` k ) = L <-> ( k =/= K /\ ( Q ` k ) = L ) ) )
24	23	rexbidv 3052	. . . . . . . . . 10 |- ( -. ( Q ` K ) = L -> ( E. k e. N ( Q ` k ) = L <-> E. k e. N ( k =/= K /\ ( Q ` k ) = L ) ) )
25		rexdifsn 4323	. . . . . . . . . 10 |- ( E. k e. ( N \ { K } ) ( Q ` k ) = L <-> E. k e. N ( k =/= K /\ ( Q ` k ) = L ) )
26	24, 25	syl6bbr 278	. . . . . . . . 9 |- ( -. ( Q ` K ) = L -> ( E. k e. N ( Q ` k ) = L <-> E. k e. ( N \ { K } ) ( Q ` k ) = L ) )
27	14, 26	syl5ibcom 235	. . . . . . . 8 |- ( ( L e. N /\ A. l e. N E. k e. N ( Q ` k ) = l ) -> ( -. ( Q ` K ) = L -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) )
28	27	ex 450	. . . . . . 7 |- ( L e. N -> ( A. l e. N E. k e. N ( Q ` k ) = l -> ( -. ( Q ` K ) = L -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) ) )
29	28	com13 88	. . . . . 6 |- ( -. ( Q ` K ) = L -> ( A. l e. N E. k e. N ( Q ` k ) = l -> ( L e. N -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) ) )
30	11, 29	syl5 34	. . . . 5 |- ( -. ( Q ` K ) = L -> ( Q e. P -> ( L e. N -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) ) )
31	9, 30	jaoi 394	. . . 4 |- ( ( -. Q e. P \/ -. ( Q ` K ) = L ) -> ( Q e. P -> ( L e. N -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) ) )
32	31	com13 88	. . 3 |- ( L e. N -> ( Q e. P -> ( ( -. Q e. P \/ -. ( Q ` K ) = L ) -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) ) )
33	32	impd 447	. 2 |- ( L e. N -> ( ( Q e. P /\ ( -. Q e. P \/ -. ( Q ` K ) = L ) ) -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) )
34	8, 33	syl5bi 232	1 |- ( L e. N -> ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> E. k e. ( N \ { K } ) ( Q ` k ) = L ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   {csn 4177   ` 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-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: (None)