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Theorem iseqf1olemqf1o 10234
Description: Lemma for seq3f1o 10245. 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 (𝜑𝐾 ∈ (𝑀...𝑁))
iseqf1olemqf.j (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1olemqf.q 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
Assertion
Ref Expression
iseqf1olemqf1o (𝜑𝑄:(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.
StepHypRef Expression
1 iseqf1olemqf.k . . . 4 (𝜑𝐾 ∈ (𝑀...𝑁))
2 iseqf1olemqf.j . . . 4 (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
3 iseqf1olemqf.q . . . 4 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
41, 2, 3iseqf1olemqf 10232 . . 3 (𝜑𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
51ad2antrr 479 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐾 ∈ (𝑀...𝑁))
62ad2antrr 479 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
7 simplrl 509 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑣 ∈ (𝑀...𝑁))
8 simplrr 510 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑤 ∈ (𝑀...𝑁))
9 simpr 109 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → (𝑄𝑣) = (𝑄𝑤))
105, 6, 3, 7, 8, 9iseqf1olemmo 10233 . . . . 5 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑣 = 𝑤)
1110ex 114 . . . 4 ((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) → ((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤))
1211ralrimivva 2491 . . 3 (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤))
13 dff13 5637 . . 3 (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ (𝑄:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤)))
144, 12, 13sylanbrc 413 . 2 (𝜑𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁))
15 elfzel1 9773 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
161, 15syl 14 . . . . 5 (𝜑𝑀 ∈ ℤ)
17 elfzel2 9772 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
181, 17syl 14 . . . . 5 (𝜑𝑁 ∈ ℤ)
1916, 18fzfigd 10172 . . . 4 (𝜑 → (𝑀...𝑁) ∈ Fin)
20 enrefg 6626 . . . 4 ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁))
2119, 20syl 14 . . 3 (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁))
22 f1finf1o 6803 . . 3 (((𝑀...𝑁) ≈ (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
2321, 19, 22syl2anc 408 . 2 (𝜑 → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
2414, 23mpbid 146 1 (𝜑𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wral 2393  ifcif 3444   class class class wbr 3899  cmpt 3959  ccnv 4508  wf 5089  1-1wf1 5090  1-1-ontowf1o 5092  cfv 5093  (class class class)co 5742  cen 6600  Fincfn 6602  1c1 7589  cmin 7901  cz 9022  ...cfz 9758
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-1o 6281  df-er 6397  df-en 6603  df-fin 6605  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-fz 9759
This theorem is referenced by:  seq3f1olemqsumkj  10239  seq3f1olemqsumk  10240  seq3f1olemstep  10242
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