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Theorem iseqf1olemqf1o 9887
Description: Lemma for seq3f1o 9898. 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 9885 . . 3 (𝜑𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
51ad2antrr 472 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐾 ∈ (𝑀...𝑁))
62ad2antrr 472 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
7 simplrl 502 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑣 ∈ (𝑀...𝑁))
8 simplrr 503 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑤 ∈ (𝑀...𝑁))
9 simpr 108 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → (𝑄𝑣) = (𝑄𝑤))
105, 6, 3, 7, 8, 9iseqf1olemmo 9886 . . . . 5 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑣 = 𝑤)
1110ex 113 . . . 4 ((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) → ((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤))
1211ralrimivva 2455 . . 3 (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤))
13 dff13 5529 . . 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 408 . 2 (𝜑𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁))
15 elfzel1 9408 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
161, 15syl 14 . . . . 5 (𝜑𝑀 ∈ ℤ)
17 elfzel2 9407 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
181, 17syl 14 . . . . 5 (𝜑𝑁 ∈ ℤ)
1916, 18fzfigd 9803 . . . 4 (𝜑 → (𝑀...𝑁) ∈ Fin)
20 enrefg 6461 . . . 4 ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁))
2119, 20syl 14 . . 3 (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁))
22 f1finf1o 6635 . . 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 403 . 2 (𝜑 → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
2414, 23mpbid 145 1 (𝜑𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  ifcif 3389   class class class wbr 3837  cmpt 3891  ccnv 4427  wf 4998  1-1wf1 4999  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  cen 6435  Fincfn 6437  1c1 7330  cmin 7632  cz 8720  ...cfz 9393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-1o 6163  df-er 6272  df-en 6438  df-fin 6440  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-fz 9394
This theorem is referenced by:  seq3f1olemqsumkj  9892  seq3f1olemqsumk  9893  seq3f1olemstep  9895
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