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Theorem iseqf1olemqf1o 10496
Description: Lemma for seq3f1o 10507. 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 10494 . . 3 (𝜑𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
51ad2antrr 488 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐾 ∈ (𝑀...𝑁))
62ad2antrr 488 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
7 simplrl 535 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑣 ∈ (𝑀...𝑁))
8 simplrr 536 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑤 ∈ (𝑀...𝑁))
9 simpr 110 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → (𝑄𝑣) = (𝑄𝑤))
105, 6, 3, 7, 8, 9iseqf1olemmo 10495 . . . . 5 (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄𝑣) = (𝑄𝑤)) → 𝑣 = 𝑤)
1110ex 115 . . . 4 ((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) → ((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤))
1211ralrimivva 2559 . . 3 (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄𝑣) = (𝑄𝑤) → 𝑣 = 𝑤))
13 dff13 5772 . . 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 417 . 2 (𝜑𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁))
15 elfzel1 10027 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
161, 15syl 14 . . . . 5 (𝜑𝑀 ∈ ℤ)
17 elfzel2 10026 . . . . . 6 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
181, 17syl 14 . . . . 5 (𝜑𝑁 ∈ ℤ)
1916, 18fzfigd 10434 . . . 4 (𝜑 → (𝑀...𝑁) ∈ Fin)
20 enrefg 6767 . . . 4 ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁))
2119, 20syl 14 . . 3 (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁))
22 f1finf1o 6949 . . 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 411 . 2 (𝜑 → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
2414, 23mpbid 147 1 (𝜑𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  ifcif 3536   class class class wbr 4005  cmpt 4066  ccnv 4627  wf 5214  1-1wf1 5215  1-1-ontowf1o 5217  cfv 5218  (class class class)co 5878  cen 6741  Fincfn 6743  1c1 7815  cmin 8131  cz 9256  ...cfz 10011
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-1o 6420  df-er 6538  df-en 6744  df-fin 6746  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-inn 8923  df-n0 9180  df-z 9257  df-uz 9532  df-fz 10012
This theorem is referenced by:  seq3f1olemqsumkj  10501  seq3f1olemqsumk  10502  seq3f1olemstep  10504
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