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Theorem frecfzennn 10206
 Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10179 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
Hypothesis
Ref Expression
frecfzennn.1 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
Assertion
Ref Expression
frecfzennn (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))

Proof of Theorem frecfzennn
Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5782 . . 3 (𝑛 = 0 → (1...𝑛) = (1...0))
2 fveq2 5421 . . 3 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
31, 2breq12d 3942 . 2 (𝑛 = 0 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...0) ≈ (𝐺‘0)))
4 oveq2 5782 . . 3 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5 fveq2 5421 . . 3 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
64, 5breq12d 3942 . 2 (𝑛 = 𝑚 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑚) ≈ (𝐺𝑚)))
7 oveq2 5782 . . 3 (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8 fveq2 5421 . . 3 (𝑛 = (𝑚 + 1) → (𝐺𝑛) = (𝐺‘(𝑚 + 1)))
97, 8breq12d 3942 . 2 (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
10 oveq2 5782 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11 fveq2 5421 . . 3 (𝑛 = 𝑁 → (𝐺𝑛) = (𝐺𝑁))
1210, 11breq12d 3942 . 2 (𝑛 = 𝑁 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑁) ≈ (𝐺𝑁)))
13 0ex 4055 . . . 4 ∅ ∈ V
1413enref 6659 . . 3 ∅ ≈ ∅
15 fz10 9833 . . 3 (1...0) = ∅
16 0zd 9073 . . . . . . 7 (⊤ → 0 ∈ ℤ)
17 frecfzennn.1 . . . . . . 7 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
1816, 17frec2uzf1od 10186 . . . . . 6 (⊤ → 𝐺:ω–1-1-onto→(ℤ‘0))
1918mptru 1340 . . . . 5 𝐺:ω–1-1-onto→(ℤ‘0)
20 peano1 4508 . . . . 5 ∅ ∈ ω
2119, 20pm3.2i 270 . . . 4 (𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω)
2216, 17frec2uz0d 10179 . . . . 5 (⊤ → (𝐺‘∅) = 0)
2322mptru 1340 . . . 4 (𝐺‘∅) = 0
24 f1ocnvfv 5680 . . . 4 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (𝐺‘0) = ∅))
2521, 23, 24mp2 16 . . 3 (𝐺‘0) = ∅
2614, 15, 253brtr4i 3958 . 2 (1...0) ≈ (𝐺‘0)
27 simpr 109 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...𝑚) ≈ (𝐺𝑚))
28 peano2nn0 9024 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
29 zex 9070 . . . . . . . . . . . . . . 15 ℤ ∈ V
3029mptex 5646 . . . . . . . . . . . . . 14 (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
31 vex 2689 . . . . . . . . . . . . . 14 𝑧 ∈ V
3230, 31fvex 5441 . . . . . . . . . . . . 13 ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
3332ax-gen 1425 . . . . . . . . . . . 12 𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
34 0z 9072 . . . . . . . . . . . 12 0 ∈ ℤ
35 frecfnom 6298 . . . . . . . . . . . 12 ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3633, 34, 35mp2an 422 . . . . . . . . . . 11 frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
3717fneq1i 5217 . . . . . . . . . . 11 (𝐺 Fn ω ↔ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3836, 37mpbir 145 . . . . . . . . . 10 𝐺 Fn ω
39 omex 4507 . . . . . . . . . 10 ω ∈ V
40 fnex 5642 . . . . . . . . . 10 ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
4138, 39, 40mp2an 422 . . . . . . . . 9 𝐺 ∈ V
4241cnvex 5077 . . . . . . . 8 𝐺 ∈ V
43 vex 2689 . . . . . . . 8 𝑚 ∈ V
4442, 43fvex 5441 . . . . . . 7 (𝐺𝑚) ∈ V
45 en2sn 6707 . . . . . . 7 (((𝑚 + 1) ∈ ℕ0 ∧ (𝐺𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
4628, 44, 45sylancl 409 . . . . . 6 (𝑚 ∈ ℕ0 → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
4746adantr 274 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
48 fzp1disj 9867 . . . . . 6 ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
4948a1i 9 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
50 f1ocnvdm 5682 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺𝑚) ∈ ω)
5119, 50mpan 420 . . . . . . . . 9 (𝑚 ∈ (ℤ‘0) → (𝐺𝑚) ∈ ω)
52 nn0uz 9367 . . . . . . . . 9 0 = (ℤ‘0)
5351, 52eleq2s 2234 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺𝑚) ∈ ω)
54 nnord 4525 . . . . . . . 8 ((𝐺𝑚) ∈ ω → Ord (𝐺𝑚))
55 ordirr 4457 . . . . . . . 8 (Ord (𝐺𝑚) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5653, 54, 553syl 17 . . . . . . 7 (𝑚 ∈ ℕ0 → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5756adantr 274 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
58 disjsn 3585 . . . . . 6 (((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅ ↔ ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5957, 58sylibr 133 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)
60 unen 6710 . . . . 5 ((((1...𝑚) ≈ (𝐺𝑚) ∧ {(𝑚 + 1)} ≈ {(𝐺𝑚)}) ∧ (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ ∧ ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
6127, 47, 49, 59, 60syl22anc 1217 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
62 1z 9087 . . . . . 6 1 ∈ ℤ
63 1m1e0 8796 . . . . . . . . . 10 (1 − 1) = 0
6463fveq2i 5424 . . . . . . . . 9 (ℤ‘(1 − 1)) = (ℤ‘0)
6552, 64eqtr4i 2163 . . . . . . . 8 0 = (ℤ‘(1 − 1))
6665eleq2i 2206 . . . . . . 7 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
6766biimpi 119 . . . . . 6 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
68 fzsuc2 9866 . . . . . 6 ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
6962, 67, 68sylancr 410 . . . . 5 (𝑚 ∈ ℕ0 → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
7069adantr 274 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
71 peano2 4509 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → suc (𝐺𝑚) ∈ ω)
7253, 71syl 14 . . . . . . . 8 (𝑚 ∈ ℕ0 → suc (𝐺𝑚) ∈ ω)
7372, 19jctil 310 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺:ω–1-1-onto→(ℤ‘0) ∧ suc (𝐺𝑚) ∈ ω))
74 0zd 9073 . . . . . . . . . 10 ((𝐺𝑚) ∈ ω → 0 ∈ ℤ)
75 id 19 . . . . . . . . . 10 ((𝐺𝑚) ∈ ω → (𝐺𝑚) ∈ ω)
7674, 17, 75frec2uzsucd 10181 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
7753, 76syl 14 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
7852eleq2i 2206 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
7978biimpi 119 . . . . . . . . . 10 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
80 f1ocnvfv2 5679 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺‘(𝐺𝑚)) = 𝑚)
8119, 79, 80sylancr 410 . . . . . . . . 9 (𝑚 ∈ ℕ0 → (𝐺‘(𝐺𝑚)) = 𝑚)
8281oveq1d 5789 . . . . . . . 8 (𝑚 ∈ ℕ0 → ((𝐺‘(𝐺𝑚)) + 1) = (𝑚 + 1))
8377, 82eqtrd 2172 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = (𝑚 + 1))
84 f1ocnvfv 5680 . . . . . . 7 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ suc (𝐺𝑚) ∈ ω) → ((𝐺‘suc (𝐺𝑚)) = (𝑚 + 1) → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚)))
8573, 83, 84sylc 62 . . . . . 6 (𝑚 ∈ ℕ0 → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚))
8685adantr 274 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚))
87 df-suc 4293 . . . . 5 suc (𝐺𝑚) = ((𝐺𝑚) ∪ {(𝐺𝑚)})
8886, 87syl6eq 2188 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = ((𝐺𝑚) ∪ {(𝐺𝑚)}))
8961, 70, 883brtr4d 3960 . . 3 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1)))
9089ex 114 . 2 (𝑚 ∈ ℕ0 → ((1...𝑚) ≈ (𝐺𝑚) → (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
913, 6, 9, 12, 26, 90nn0ind 9172 1 (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1329   = wceq 1331  ⊤wtru 1332   ∈ wcel 1480  Vcvv 2686   ∪ cun 3069   ∩ cin 3070  ∅c0 3363  {csn 3527   class class class wbr 3929   ↦ cmpt 3989  Ord word 4284  suc csuc 4287  ωcom 4504  ◡ccnv 4538   Fn wfn 5118  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774  freccfrec 6287   ≈ cen 6632  0cc0 7627  1c1 7628   + caddc 7630   − cmin 7940  ℕ0cn0 8984  ℤcz 9061  ℤ≥cuz 9333  ...cfz 9797 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743 This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-1o 6313  df-er 6429  df-en 6635  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798 This theorem is referenced by:  frecfzen2  10207  hashfz1  10536
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