Theorem iseqf1olemqf1o 10872

Description: Lemma for seq3f1o 10883. 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 10870	. . 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 537	. . . . . 6 |- ( ( ( ph /\ ( v e. ( M ... N ) /\ w e. ( M ... N ) ) ) /\ ( Q ` v ) = ( Q ` w ) ) -> v e. ( M ... N ) )
8		simplrr 538	. . . . . 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 10871	. . . . 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 2626	. . 3 |- ( ph -> A. v e. ( M ... N ) A. w e. ( M ... N ) ( ( Q ` v ) = ( Q ` w ) -> v = w ) )
13		dff13 5943	. . 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 10361	. . . . . 6 |- ( K e. ( M ... N ) -> M e. ZZ )
16	1, 15	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
17		elfzel2 10360	. . . . . 6 |- ( K e. ( M ... N ) -> N e. ZZ )
18	1, 17	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
19	16, 18	fzfigd 10797	. . . 4 |- ( ph -> ( M ... N ) e. Fin )
20		enrefg 7005	. . . 4 |- ( ( M ... N ) e. Fin -> ( M ... N ) ~~ ( M ... N ) )
21	19, 20	syl 14	. . 3 |- ( ph -> ( M ... N ) ~~ ( M ... N ) )
22		f1finf1o 7219	. . 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 1398   e. wcel 2205   A.wral 2522   ifcif 3622   class class class wbr 4111   |-> cmpt 4173   `'ccnv 4750   -->wf 5350   -1-1->wf1 5351   -1-1-onto->wf1o 5353   ` cfv 5354  (class class class)co 6052   ~~ cen 6975   Fincfn 6977   1c1 8130   - cmin 8446   ZZcz 9579   ...cfz 10345

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-en 6978  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346

This theorem is referenced by:  seq3f1olemqsumkj  10877  seq3f1olemqsumk  10878  seq3f1olemstep  10880