Theorem iseqf1olemqf1o 10615

Description: Lemma for seq3f1o 10626. 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 10613	. . 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 10614	. . . . 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 2579	. . 3 |- ( ph -> A. v e. ( M ... N ) A. w e. ( M ... N ) ( ( Q ` v ) = ( Q ` w ) -> v = w ) )
13		dff13 5818	. . 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 10116	. . . . . 6 |- ( K e. ( M ... N ) -> M e. ZZ )
16	1, 15	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
17		elfzel2 10115	. . . . . 6 |- ( K e. ( M ... N ) -> N e. ZZ )
18	1, 17	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
19	16, 18	fzfigd 10540	. . . 4 |- ( ph -> ( M ... N ) e. Fin )
20		enrefg 6832	. . . 4 |- ( ( M ... N ) e. Fin -> ( M ... N ) ~~ ( M ... N ) )
21	19, 20	syl 14	. . 3 |- ( ph -> ( M ... N ) ~~ ( M ... N ) )
22		f1finf1o 7022	. . 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 2167   A.wral 2475   ifcif 3562   class class class wbr 4034   |-> cmpt 4095   `'ccnv 4663   -->wf 5255   -1-1->wf1 5256   -1-1-onto->wf1o 5258   ` cfv 5259  (class class class)co 5925   ~~ cen 6806   Fincfn 6808   1c1 7897   - cmin 8214   ZZcz 9343   ...cfz 10100

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101

This theorem is referenced by:  seq3f1olemqsumkj  10620  seq3f1olemqsumk  10621  seq3f1olemstep  10623