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Theorem iseqf1olemqf1o 10217
Description: Lemma for seq3f1o 10228. 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 10215 . . 3 (𝜑𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
51ad2antrr 477 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐾 ∈ (𝑀...𝑁))
62ad2antrr 477 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
7 simplrl 507 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑣 ∈ (𝑀...𝑁))
8 simplrr 508 . . . . . 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 10216 . . . . 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 2489 . . 3 (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤))
13 dff13 5635 . . 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 411 . 2 (𝜑𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁))
15 elfzel1 9756 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
161, 15syl 14 . . . . 5 (𝜑𝑀 ∈ ℤ)
17 elfzel2 9755 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
181, 17syl 14 . . . . 5 (𝜑𝑁 ∈ ℤ)
1916, 18fzfigd 10155 . . . 4 (𝜑 → (𝑀...𝑁) ∈ Fin)
20 enrefg 6624 . . . 4 ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁))
2119, 20syl 14 . . 3 (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁))
22 f1finf1o 6801 . . 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 406 . 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 1314  wcel 1463  wral 2391  ifcif 3442   class class class wbr 3897  cmpt 3957  ccnv 4506  wf 5087  1-1wf1 5088  1-1-ontowf1o 5090  cfv 5091  (class class class)co 5740  cen 6598  Fincfn 6600  1c1 7585  cmin 7897  cz 9008  ...cfz 9741
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-1o 6279  df-er 6395  df-en 6601  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-fz 9742
This theorem is referenced by:  seq3f1olemqsumkj  10222  seq3f1olemqsumk  10223  seq3f1olemstep  10225
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