Theorem iseqf1olemqf1o 10495

Description: Lemma for seq3f1o 10506. 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 10493	. . 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 10494	. . . . 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 2559	. . 3 |- ( ph -> A. v e. ( M ... N ) A. w e. ( M ... N ) ( ( Q ` v ) = ( Q ` w ) -> v = w ) )
13		dff13 5771	. . 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 10026	. . . . . 6 |- ( K e. ( M ... N ) -> M e. ZZ )
16	1, 15	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
17		elfzel2 10025	. . . . . 6 |- ( K e. ( M ... N ) -> N e. ZZ )
18	1, 17	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
19	16, 18	fzfigd 10433	. . . 4 |- ( ph -> ( M ... N ) e. Fin )
20		enrefg 6766	. . . 4 |- ( ( M ... N ) e. Fin -> ( M ... N ) ~~ ( M ... N ) )
21	19, 20	syl 14	. . 3 |- ( ph -> ( M ... N ) ~~ ( M ... N ) )
22		f1finf1o 6948	. . 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 1353   e. wcel 2148   A.wral 2455   ifcif 3536   class class class wbr 4005   |-> cmpt 4066   `'ccnv 4627   -->wf 5214   -1-1->wf1 5215   -1-1-onto->wf1o 5217   ` cfv 5218  (class class class)co 5877   ~~ cen 6740   Fincfn 6742   1c1 7814   - cmin 8130   ZZcz 9255   ...cfz 10010

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-1o 6419  df-er 6537  df-en 6743  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011

This theorem is referenced by:  seq3f1olemqsumkj  10500  seq3f1olemqsumk  10501  seq3f1olemstep  10503