Theorem iseqf1olemqf1o 10688

Description: Lemma for seq3f1o 10699. 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 10686	. . 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 535	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝑣 ∈ (𝑀...𝑁))
8		simplrr 536	. . . . . 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 10687	. . . . 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 2590	. . 3 ⊢ (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄‘𝑣) = (𝑄‘𝑤) → 𝑣 = 𝑤))
13		dff13 5860	. . 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 10181	. . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
16	1, 15	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
17		elfzel2 10180	. . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
18	1, 17	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
19	16, 18	fzfigd 10613	. . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
20		enrefg 6878	. . . 4 ⊢ ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁))
21	19, 20	syl 14	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁))
22		f1finf1o 7075	. . 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 1373   ∈ wcel 2178  ∀wral 2486  ifcif 3579   class class class wbr 4059   ↦ cmpt 4121  ^◡ccnv 4692  ⟶wf 5286  –1-1→wf1 5287  –1-1-onto→wf1o 5289  ‘cfv 5290  (class class class)co 5967   ≈ cen 6848  Fincfn 6850  1c1 7961   − cmin 8278  ℤcz 9407  ...cfz 10165

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-er 6643  df-en 6851  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166

This theorem is referenced by:  seq3f1olemqsumkj  10693  seq3f1olemqsumk  10694  seq3f1olemstep  10696