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Theorem frecfzennn 10325
Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10298 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 5832 . . 3 (𝑛 = 0 → (1...𝑛) = (1...0))
2 fveq2 5468 . . 3 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
31, 2breq12d 3978 . 2 (𝑛 = 0 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...0) ≈ (𝐺‘0)))
4 oveq2 5832 . . 3 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5 fveq2 5468 . . 3 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
64, 5breq12d 3978 . 2 (𝑛 = 𝑚 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑚) ≈ (𝐺𝑚)))
7 oveq2 5832 . . 3 (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8 fveq2 5468 . . 3 (𝑛 = (𝑚 + 1) → (𝐺𝑛) = (𝐺‘(𝑚 + 1)))
97, 8breq12d 3978 . 2 (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
10 oveq2 5832 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11 fveq2 5468 . . 3 (𝑛 = 𝑁 → (𝐺𝑛) = (𝐺𝑁))
1210, 11breq12d 3978 . 2 (𝑛 = 𝑁 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑁) ≈ (𝐺𝑁)))
13 0ex 4091 . . . 4 ∅ ∈ V
1413enref 6710 . . 3 ∅ ≈ ∅
15 fz10 9948 . . 3 (1...0) = ∅
16 0zd 9179 . . . . . . 7 (⊤ → 0 ∈ ℤ)
17 frecfzennn.1 . . . . . . 7 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
1816, 17frec2uzf1od 10305 . . . . . 6 (⊤ → 𝐺:ω–1-1-onto→(ℤ‘0))
1918mptru 1344 . . . . 5 𝐺:ω–1-1-onto→(ℤ‘0)
20 peano1 4553 . . . . 5 ∅ ∈ ω
2119, 20pm3.2i 270 . . . 4 (𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω)
2216, 17frec2uz0d 10298 . . . . 5 (⊤ → (𝐺‘∅) = 0)
2322mptru 1344 . . . 4 (𝐺‘∅) = 0
24 f1ocnvfv 5729 . . . 4 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (𝐺‘0) = ∅))
2521, 23, 24mp2 16 . . 3 (𝐺‘0) = ∅
2614, 15, 253brtr4i 3994 . 2 (1...0) ≈ (𝐺‘0)
27 simpr 109 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...𝑚) ≈ (𝐺𝑚))
28 peano2nn0 9130 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
29 zex 9176 . . . . . . . . . . . . . . 15 ℤ ∈ V
3029mptex 5693 . . . . . . . . . . . . . 14 (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
31 vex 2715 . . . . . . . . . . . . . 14 𝑧 ∈ V
3230, 31fvex 5488 . . . . . . . . . . . . 13 ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
3332ax-gen 1429 . . . . . . . . . . . 12 𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
34 0z 9178 . . . . . . . . . . . 12 0 ∈ ℤ
35 frecfnom 6348 . . . . . . . . . . . 12 ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3633, 34, 35mp2an 423 . . . . . . . . . . 11 frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
3717fneq1i 5264 . . . . . . . . . . 11 (𝐺 Fn ω ↔ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3836, 37mpbir 145 . . . . . . . . . 10 𝐺 Fn ω
39 omex 4552 . . . . . . . . . 10 ω ∈ V
40 fnex 5689 . . . . . . . . . 10 ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
4138, 39, 40mp2an 423 . . . . . . . . 9 𝐺 ∈ V
4241cnvex 5124 . . . . . . . 8 𝐺 ∈ V
43 vex 2715 . . . . . . . 8 𝑚 ∈ V
4442, 43fvex 5488 . . . . . . 7 (𝐺𝑚) ∈ V
45 en2sn 6758 . . . . . . 7 (((𝑚 + 1) ∈ ℕ0 ∧ (𝐺𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
4628, 44, 45sylancl 410 . . . . . 6 (𝑚 ∈ ℕ0 → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
4746adantr 274 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
48 fzp1disj 9982 . . . . . 6 ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
4948a1i 9 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
50 f1ocnvdm 5731 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺𝑚) ∈ ω)
5119, 50mpan 421 . . . . . . . . 9 (𝑚 ∈ (ℤ‘0) → (𝐺𝑚) ∈ ω)
52 nn0uz 9473 . . . . . . . . 9 0 = (ℤ‘0)
5351, 52eleq2s 2252 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺𝑚) ∈ ω)
54 nnord 4571 . . . . . . . 8 ((𝐺𝑚) ∈ ω → Ord (𝐺𝑚))
55 ordirr 4501 . . . . . . . 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 3621 . . . . . 6 (((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅ ↔ ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5957, 58sylibr 133 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)
60 unen 6761 . . . . 5 ((((1...𝑚) ≈ (𝐺𝑚) ∧ {(𝑚 + 1)} ≈ {(𝐺𝑚)}) ∧ (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ ∧ ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
6127, 47, 49, 59, 60syl22anc 1221 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
62 1z 9193 . . . . . 6 1 ∈ ℤ
63 1m1e0 8902 . . . . . . . . . 10 (1 − 1) = 0
6463fveq2i 5471 . . . . . . . . 9 (ℤ‘(1 − 1)) = (ℤ‘0)
6552, 64eqtr4i 2181 . . . . . . . 8 0 = (ℤ‘(1 − 1))
6665eleq2i 2224 . . . . . . 7 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
6766biimpi 119 . . . . . 6 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
68 fzsuc2 9981 . . . . . 6 ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
6962, 67, 68sylancr 411 . . . . 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 4554 . . . . . . . . 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 9179 . . . . . . . . . 10 ((𝐺𝑚) ∈ ω → 0 ∈ ℤ)
75 id 19 . . . . . . . . . 10 ((𝐺𝑚) ∈ ω → (𝐺𝑚) ∈ ω)
7674, 17, 75frec2uzsucd 10300 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
7753, 76syl 14 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
7852eleq2i 2224 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
7978biimpi 119 . . . . . . . . . 10 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
80 f1ocnvfv2 5728 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺‘(𝐺𝑚)) = 𝑚)
8119, 79, 80sylancr 411 . . . . . . . . 9 (𝑚 ∈ ℕ0 → (𝐺‘(𝐺𝑚)) = 𝑚)
8281oveq1d 5839 . . . . . . . 8 (𝑚 ∈ ℕ0 → ((𝐺‘(𝐺𝑚)) + 1) = (𝑚 + 1))
8377, 82eqtrd 2190 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = (𝑚 + 1))
84 f1ocnvfv 5729 . . . . . . 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 4331 . . . . 5 suc (𝐺𝑚) = ((𝐺𝑚) ∪ {(𝐺𝑚)})
8886, 87eqtrdi 2206 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = ((𝐺𝑚) ∪ {(𝐺𝑚)}))
8961, 70, 883brtr4d 3996 . . 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 9278 1 (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wal 1333   = wceq 1335  wtru 1336  wcel 2128  Vcvv 2712  cun 3100  cin 3101  c0 3394  {csn 3560   class class class wbr 3965  cmpt 4025  Ord word 4322  suc csuc 4325  ωcom 4549  ccnv 4585   Fn wfn 5165  1-1-ontowf1o 5169  cfv 5170  (class class class)co 5824  freccfrec 6337  cen 6683  0cc0 7732  1c1 7733   + caddc 7735  cmin 8046  0cn0 9090  cz 9167  cuz 9439  ...cfz 9912
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-recs 6252  df-frec 6338  df-1o 6363  df-er 6480  df-en 6686  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-inn 8834  df-n0 9091  df-z 9168  df-uz 9440  df-fz 9913
This theorem is referenced by:  frecfzen2  10326  hashfz1  10657
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