Theorem iseqf1olemqf1o 10436

Description: Lemma for seq3f1o 10447. 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 10434	. . 3 |- ( ph -> Q : ( M ... N ) --> ( M ... N ) )
5	1	ad2antrr 485	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> K e. ( M ... N ) )
6	2	ad2antrr 485	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7		simplrl 530	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> v e. ( M ... N ) )
8		simplrr 531	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> w e. ( M ... N ) )
9		simpr 109	. . . . . 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 10435	. . . . 5 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> v = w )
11	10	ex 114	. . . 4 |- ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) -> ( ( Q ` v ) = ( Q ` w ) -> v = w ) )
12	11	ralrimivva 2552	. . 3 |- ( ph -> A. v e. ( M ... N ) A. w e. ( M ... N ) ( ( Q ` v ) = ( Q ` w ) -> v = w ) )
13		dff13 5744	. . 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 415	. 2 |- ( ph -> Q : ( M ... N ) -1-1-> ( M ... N ) )
15		elfzel1 9967	. . . . . 6 |- ( K e. ( M ... N ) -> M e. ZZ )
16	1, 15	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
17		elfzel2 9966	. . . . . 6 |- ( K e. ( M ... N ) -> N e. ZZ )
18	1, 17	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
19	16, 18	fzfigd 10374	. . . 4 |- ( ph -> ( M ... N ) e. Fin )
20		enrefg 6738	. . . 4 |- ( ( M ... N ) e. Fin -> ( M ... N ) ~~ ( M ... N ) )
21	19, 20	syl 14	. . 3 |- ( ph -> ( M ... N ) ~~ ( M ... N ) )
22		f1finf1o 6920	. . 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 409	. 2 |- ( ph -> ( Q : ( M ... N ) -1-1-> ( M ... N ) <-> Q : ( M ... N ) -1-1-onto-> ( M ... N ) ) )
24	14, 23	mpbid 146	1 |- ( ph -> Q : ( M ... N ) -1-1-onto-> ( M ... N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   ifcif 3525   class class class wbr 3987   |-> cmpt 4048   `'ccnv 4608   -->wf 5192   -1-1->wf1 5193   -1-1-onto->wf1o 5195   ` cfv 5196  (class class class)co 5850   ~~ cen 6712   Fincfn 6714   1c1 7762   - cmin 8077   ZZcz 9199   ...cfz 9952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-1o 6392  df-er 6509  df-en 6715  df-fin 6717  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-fz 9953

This theorem is referenced by:  seq3f1olemqsumkj  10441  seq3f1olemqsumk  10442  seq3f1olemstep  10444