Theorem iseqf1olemqf1o 10868

Description: Lemma for seq3f1o 10879. 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	⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
iseqf1olemqf.j	⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1olemqf.q	⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))

Assertion

Ref	Expression
iseqf1olemqf1o	⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))

Distinct variable groups:   𝑢,𝐽   𝑢,𝐾   𝑢,𝑀   𝑢,𝑁   𝜑,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 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
2		iseqf1olemqf.j	. . . 4 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
3		iseqf1olemqf.q	. . . 4 ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))
4	1, 2, 3	iseqf1olemqf 10866	. . 3 ⊢ (𝜑 → 𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
5	1	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝐾 ∈ (𝑀...𝑁))
6	2	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
7		simplrl 537	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝑣 ∈ (𝑀...𝑁))
8		simplrr 538	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝑤 ∈ (𝑀...𝑁))
9		simpr 110	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → (𝑄‘𝑣) = (𝑄‘𝑤))
10	5, 6, 3, 7, 8, 9	iseqf1olemmo 10867	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝑣 = 𝑤)
11	10	ex 115	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) → ((𝑄‘𝑣) = (𝑄‘𝑤) → 𝑣 = 𝑤))
12	11	ralrimivva 2624	. . 3 ⊢ (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄‘𝑣) = (𝑄‘𝑤) → 𝑣 = 𝑤))
13		dff13 5941	. . 3 ⊢ (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ (𝑄:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄‘𝑣) = (𝑄‘𝑤) → 𝑣 = 𝑤)))
14	4, 12, 13	sylanbrc 417	. 2 ⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁))
15		elfzel1 10358	. . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
16	1, 15	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
17		elfzel2 10357	. . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
18	1, 17	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
19	16, 18	fzfigd 10793	. . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
20		enrefg 7003	. . . 4 ⊢ ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁))
21	19, 20	syl 14	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁))
22		f1finf1o 7217	. . 3 ⊢ (((𝑀...𝑁) ≈ (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
23	21, 19, 22	syl2anc 411	. 2 ⊢ (𝜑 → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
24	14, 23	mpbid 147	1 ⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ifcif 3620   class class class wbr 4109   ↦ cmpt 4171  ^◡ccnv 4748  ⟶wf 5348  –1-1→wf1 5349  –1-1-onto→wf1o 5351  ‘cfv 5352  (class class class)co 6050   ≈ cen 6973  Fincfn 6975  1c1 8128   − cmin 8444  ℤcz 9577  ...cfz 10342

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343

This theorem is referenced by:  seq3f1olemqsumkj  10873  seq3f1olemqsumk  10874  seq3f1olemstep  10876