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Theorem frecfzennn 9894
Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 9867 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 5674 . . 3 (𝑛 = 0 → (1...𝑛) = (1...0))
2 fveq2 5318 . . 3 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
31, 2breq12d 3864 . 2 (𝑛 = 0 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...0) ≈ (𝐺‘0)))
4 oveq2 5674 . . 3 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5 fveq2 5318 . . 3 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
64, 5breq12d 3864 . 2 (𝑛 = 𝑚 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑚) ≈ (𝐺𝑚)))
7 oveq2 5674 . . 3 (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8 fveq2 5318 . . 3 (𝑛 = (𝑚 + 1) → (𝐺𝑛) = (𝐺‘(𝑚 + 1)))
97, 8breq12d 3864 . 2 (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
10 oveq2 5674 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11 fveq2 5318 . . 3 (𝑛 = 𝑁 → (𝐺𝑛) = (𝐺𝑁))
1210, 11breq12d 3864 . 2 (𝑛 = 𝑁 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑁) ≈ (𝐺𝑁)))
13 0ex 3972 . . . 4 ∅ ∈ V
1413enref 6536 . . 3 ∅ ≈ ∅
15 fz10 9521 . . 3 (1...0) = ∅
16 0zd 8823 . . . . . . 7 (⊤ → 0 ∈ ℤ)
17 frecfzennn.1 . . . . . . 7 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
1816, 17frec2uzf1od 9874 . . . . . 6 (⊤ → 𝐺:ω–1-1-onto→(ℤ‘0))
1918mptru 1299 . . . . 5 𝐺:ω–1-1-onto→(ℤ‘0)
20 peano1 4422 . . . . 5 ∅ ∈ ω
2119, 20pm3.2i 267 . . . 4 (𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω)
2216, 17frec2uz0d 9867 . . . . 5 (⊤ → (𝐺‘∅) = 0)
2322mptru 1299 . . . 4 (𝐺‘∅) = 0
24 f1ocnvfv 5572 . . . 4 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (𝐺‘0) = ∅))
2521, 23, 24mp2 16 . . 3 (𝐺‘0) = ∅
2614, 15, 253brtr4i 3879 . 2 (1...0) ≈ (𝐺‘0)
27 simpr 109 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...𝑚) ≈ (𝐺𝑚))
28 peano2nn0 8774 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
29 zex 8820 . . . . . . . . . . . . . . 15 ℤ ∈ V
3029mptex 5537 . . . . . . . . . . . . . 14 (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
31 vex 2623 . . . . . . . . . . . . . 14 𝑧 ∈ V
3230, 31fvex 5338 . . . . . . . . . . . . 13 ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
3332ax-gen 1384 . . . . . . . . . . . 12 𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
34 0z 8822 . . . . . . . . . . . 12 0 ∈ ℤ
35 frecfnom 6180 . . . . . . . . . . . 12 ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3633, 34, 35mp2an 418 . . . . . . . . . . 11 frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
3717fneq1i 5121 . . . . . . . . . . 11 (𝐺 Fn ω ↔ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3836, 37mpbir 145 . . . . . . . . . 10 𝐺 Fn ω
39 omex 4421 . . . . . . . . . 10 ω ∈ V
40 fnex 5533 . . . . . . . . . 10 ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
4138, 39, 40mp2an 418 . . . . . . . . 9 𝐺 ∈ V
4241cnvex 4982 . . . . . . . 8 𝐺 ∈ V
43 vex 2623 . . . . . . . 8 𝑚 ∈ V
4442, 43fvex 5338 . . . . . . 7 (𝐺𝑚) ∈ V
45 en2sn 6584 . . . . . . 7 (((𝑚 + 1) ∈ ℕ0 ∧ (𝐺𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
4628, 44, 45sylancl 405 . . . . . 6 (𝑚 ∈ ℕ0 → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
4746adantr 271 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
48 fzp1disj 9555 . . . . . 6 ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
4948a1i 9 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
50 f1ocnvdm 5574 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺𝑚) ∈ ω)
5119, 50mpan 416 . . . . . . . . 9 (𝑚 ∈ (ℤ‘0) → (𝐺𝑚) ∈ ω)
52 nn0uz 9114 . . . . . . . . 9 0 = (ℤ‘0)
5351, 52eleq2s 2183 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺𝑚) ∈ ω)
54 nnord 4439 . . . . . . . 8 ((𝐺𝑚) ∈ ω → Ord (𝐺𝑚))
55 ordirr 4371 . . . . . . . 8 (Ord (𝐺𝑚) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5653, 54, 553syl 17 . . . . . . 7 (𝑚 ∈ ℕ0 → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5756adantr 271 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
58 disjsn 3508 . . . . . 6 (((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅ ↔ ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5957, 58sylibr 133 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)
60 unen 6587 . . . . 5 ((((1...𝑚) ≈ (𝐺𝑚) ∧ {(𝑚 + 1)} ≈ {(𝐺𝑚)}) ∧ (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ ∧ ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
6127, 47, 49, 59, 60syl22anc 1176 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
62 1z 8837 . . . . . 6 1 ∈ ℤ
63 1m1e0 8552 . . . . . . . . . 10 (1 − 1) = 0
6463fveq2i 5321 . . . . . . . . 9 (ℤ‘(1 − 1)) = (ℤ‘0)
6552, 64eqtr4i 2112 . . . . . . . 8 0 = (ℤ‘(1 − 1))
6665eleq2i 2155 . . . . . . 7 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
6766biimpi 119 . . . . . 6 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
68 fzsuc2 9554 . . . . . 6 ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
6962, 67, 68sylancr 406 . . . . 5 (𝑚 ∈ ℕ0 → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
7069adantr 271 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
71 peano2 4423 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → suc (𝐺𝑚) ∈ ω)
7253, 71syl 14 . . . . . . . 8 (𝑚 ∈ ℕ0 → suc (𝐺𝑚) ∈ ω)
7372, 19jctil 306 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺:ω–1-1-onto→(ℤ‘0) ∧ suc (𝐺𝑚) ∈ ω))
74 0zd 8823 . . . . . . . . . 10 ((𝐺𝑚) ∈ ω → 0 ∈ ℤ)
75 id 19 . . . . . . . . . 10 ((𝐺𝑚) ∈ ω → (𝐺𝑚) ∈ ω)
7674, 17, 75frec2uzsucd 9869 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
7753, 76syl 14 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
7852eleq2i 2155 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
7978biimpi 119 . . . . . . . . . 10 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
80 f1ocnvfv2 5571 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺‘(𝐺𝑚)) = 𝑚)
8119, 79, 80sylancr 406 . . . . . . . . 9 (𝑚 ∈ ℕ0 → (𝐺‘(𝐺𝑚)) = 𝑚)
8281oveq1d 5681 . . . . . . . 8 (𝑚 ∈ ℕ0 → ((𝐺‘(𝐺𝑚)) + 1) = (𝑚 + 1))
8377, 82eqtrd 2121 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = (𝑚 + 1))
84 f1ocnvfv 5572 . . . . . . 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 271 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚))
87 df-suc 4207 . . . . 5 suc (𝐺𝑚) = ((𝐺𝑚) ∪ {(𝐺𝑚)})
8886, 87syl6eq 2137 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = ((𝐺𝑚) ∪ {(𝐺𝑚)}))
8961, 70, 883brtr4d 3881 . . 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 8921 1 (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wal 1288   = wceq 1290  wtru 1291  wcel 1439  Vcvv 2620  cun 2998  cin 2999  c0 3287  {csn 3450   class class class wbr 3851  cmpt 3905  Ord word 4198  suc csuc 4201  ωcom 4418  ccnv 4451   Fn wfn 5023  1-1-ontowf1o 5027  cfv 5028  (class class class)co 5666  freccfrec 6169  cen 6509  0cc0 7411  1c1 7412   + caddc 7414  cmin 7714  0cn0 8734  cz 8811  cuz 9080  ...cfz 9485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-recs 6084  df-frec 6170  df-1o 6195  df-er 6306  df-en 6512  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-n0 8735  df-z 8812  df-uz 9081  df-fz 9486
This theorem is referenced by:  frecfzen2  9895  hashfz1  10252
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