Theorem iseqf1olemqf1o 10580

Description: Lemma for seq3f1o 10591. Q is a permutation of ( M ... N ). Q is formed from the constant portion of J, followed by the single element K (at position K), followed by the rest of J (with the K deleted and the elements before K moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.)

Hypotheses

Ref	Expression
iseqf1olemqf.k	|- ( ph -> K e. ( M ... N ) )
iseqf1olemqf.j	|- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemqf.q	|- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )

Assertion

Ref	Expression
iseqf1olemqf1o	|- ( ph -> Q : ( M ... N ) -1-1-onto-> ( M ... N ) )

Distinct variable groups:   u, J   u, K   u, M   u, N   ph, u

Allowed substitution hint:   Q( u)

Proof of Theorem iseqf1olemqf1o

Dummy variables v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqf1olemqf.k	. . . 4 |- ( ph -> K e. ( M ... N ) )
2		iseqf1olemqf.j	. . . 4 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
3		iseqf1olemqf.q	. . . 4 |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )
4	1, 2, 3	iseqf1olemqf 10578	. . 3 |- ( ph -> Q : ( M ... N ) --> ( M ... N ) )
5	1	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> K e. ( M ... N ) )
6	2	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7		simplrl 535	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> v e. ( M ... N ) )
8		simplrr 536	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> w e. ( M ... N ) )
9		simpr 110	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> ( Q ` v ) = ( Q ` w ) )
10	5, 6, 3, 7, 8, 9	iseqf1olemmo 10579	. . . . 5 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> v = w )
11	10	ex 115	. . . 4 |- ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) -> ( ( Q ` v ) = ( Q ` w ) -> v = w ) )
12	11	ralrimivva 2576	. . 3 |- ( ph -> A. v e. ( M ... N ) A. w e. ( M ... N ) ( ( Q ` v ) = ( Q ` w ) -> v = w ) )
13		dff13 5812	. . 3 |- ( Q : ( M ... N ) -1-1-> ( M ... N ) <-> ( Q : ( M ... N ) --> ( M ... N ) /\ A. v e. ( M ... N ) A. w e. ( M ... N ) ( ( Q ` v ) = ( Q ` w ) -> v = w ) ) )
14	4, 12, 13	sylanbrc 417	. 2 |- ( ph -> Q : ( M ... N ) -1-1-> ( M ... N ) )
15		elfzel1 10093	. . . . . 6 |- ( K e. ( M ... N ) -> M e. ZZ )
16	1, 15	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
17		elfzel2 10092	. . . . . 6 |- ( K e. ( M ... N ) -> N e. ZZ )
18	1, 17	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
19	16, 18	fzfigd 10505	. . . 4 |- ( ph -> ( M ... N ) e. Fin )
20		enrefg 6820	. . . 4 |- ( ( M ... N ) e. Fin -> ( M ... N ) ~~ ( M ... N ) )
21	19, 20	syl 14	. . 3 |- ( ph -> ( M ... N ) ~~ ( M ... N ) )
22		f1finf1o 7008	. . 3 |- ( ( ( M ... N ) ~~ ( M ... N ) /\ ( M ... N ) e. Fin ) -> ( Q : ( M ... N ) -1-1-> ( M ... N ) <-> Q : ( M ... N ) -1-1-onto-> ( M ... N ) ) )
23	21, 19, 22	syl2anc 411	. 2 |- ( ph -> ( Q : ( M ... N ) -1-1-> ( M ... N ) <-> Q : ( M ... N ) -1-1-onto-> ( M ... N ) ) )
24	14, 23	mpbid 147	1 |- ( ph -> Q : ( M ... N ) -1-1-onto-> ( M ... N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   ifcif 3558   class class class wbr 4030   |-> cmpt 4091   `'ccnv 4659   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5919   ~~ cen 6794   Fincfn 6796   1c1 7875   - cmin 8192   ZZcz 9320   ...cfz 10077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-1o 6471  df-er 6589  df-en 6797  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078

This theorem is referenced by:  seq3f1olemqsumkj  10585  seq3f1olemqsumk  10586  seq3f1olemstep  10588