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Theorem f1ocnt 30451
Description: Given a countable set 𝐴, number its elements by providing a one-to-one mapping either with or an integer range starting from 1. The domain of the function can then be used with iundisjcnt 30447 or iundisj2cnt 30448. (Contributed by Thierry Arnoux, 25-Jul-2020.)
Assertion
Ref Expression
f1ocnt (𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
Distinct variable group:   𝐴,𝑓

Proof of Theorem f1ocnt
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 f1o0 6644 . . . . . . 7 ∅:∅–1-1-onto→∅
2 eqidd 2819 . . . . . . . 8 (𝐴 = ∅ → ∅ = ∅)
3 dm0 5783 . . . . . . . . 9 dom ∅ = ∅
43a1i 11 . . . . . . . 8 (𝐴 = ∅ → dom ∅ = ∅)
5 id 22 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
62, 4, 5f1oeq123d 6603 . . . . . . 7 (𝐴 = ∅ → (∅:dom ∅–1-1-onto𝐴 ↔ ∅:∅–1-1-onto→∅))
71, 6mpbiri 259 . . . . . 6 (𝐴 = ∅ → ∅:dom ∅–1-1-onto𝐴)
8 fveq2 6663 . . . . . . . . . . . . 13 (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
9 hash0 13716 . . . . . . . . . . . . 13 (♯‘∅) = 0
108, 9syl6eq 2869 . . . . . . . . . . . 12 (𝐴 = ∅ → (♯‘𝐴) = 0)
1110oveq1d 7160 . . . . . . . . . . 11 (𝐴 = ∅ → ((♯‘𝐴) + 1) = (0 + 1))
12 0p1e1 11747 . . . . . . . . . . 11 (0 + 1) = 1
1311, 12syl6eq 2869 . . . . . . . . . 10 (𝐴 = ∅ → ((♯‘𝐴) + 1) = 1)
1413oveq2d 7161 . . . . . . . . 9 (𝐴 = ∅ → (1..^((♯‘𝐴) + 1)) = (1..^1))
15 fzo0 13049 . . . . . . . . 9 (1..^1) = ∅
1614, 15syl6eq 2869 . . . . . . . 8 (𝐴 = ∅ → (1..^((♯‘𝐴) + 1)) = ∅)
174, 16eqtr4d 2856 . . . . . . 7 (𝐴 = ∅ → dom ∅ = (1..^((♯‘𝐴) + 1)))
1817olcd 870 . . . . . 6 (𝐴 = ∅ → (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1))))
197, 18jca 512 . . . . 5 (𝐴 = ∅ → (∅:dom ∅–1-1-onto𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))))
20 0ex 5202 . . . . . 6 ∅ ∈ V
21 id 22 . . . . . . . 8 (𝑓 = ∅ → 𝑓 = ∅)
22 dmeq 5765 . . . . . . . 8 (𝑓 = ∅ → dom 𝑓 = dom ∅)
23 eqidd 2819 . . . . . . . 8 (𝑓 = ∅ → 𝐴 = 𝐴)
2421, 22, 23f1oeq123d 6603 . . . . . . 7 (𝑓 = ∅ → (𝑓:dom 𝑓1-1-onto𝐴 ↔ ∅:dom ∅–1-1-onto𝐴))
2522eqeq1d 2820 . . . . . . . 8 (𝑓 = ∅ → (dom 𝑓 = ℕ ↔ dom ∅ = ℕ))
2622eqeq1d 2820 . . . . . . . 8 (𝑓 = ∅ → (dom 𝑓 = (1..^((♯‘𝐴) + 1)) ↔ dom ∅ = (1..^((♯‘𝐴) + 1))))
2725, 26orbi12d 912 . . . . . . 7 (𝑓 = ∅ → ((dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))) ↔ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))))
2824, 27anbi12d 630 . . . . . 6 (𝑓 = ∅ → ((𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))) ↔ (∅:dom ∅–1-1-onto𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1))))))
2920, 28spcev 3604 . . . . 5 ((∅:dom ∅–1-1-onto𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))) → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
3019, 29syl 17 . . . 4 (𝐴 = ∅ → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
3130adantl 482 . . 3 (((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) ∧ 𝐴 = ∅) → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
32 f1odm 6612 . . . . . . . . . . 11 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
3332f1oeq2d 6604 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝑓:dom 𝑓1-1-onto𝐴𝑓:(1...(♯‘𝐴))–1-1-onto𝐴))
3433ibir 269 . . . . . . . . 9 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:dom 𝑓1-1-onto𝐴)
3534adantl 482 . . . . . . . 8 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑓:dom 𝑓1-1-onto𝐴)
3632adantl 482 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → dom 𝑓 = (1...(♯‘𝐴)))
37 simpl 483 . . . . . . . . . . . 12 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (♯‘𝐴) ∈ ℕ)
3837nnzd 12074 . . . . . . . . . . 11 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (♯‘𝐴) ∈ ℤ)
39 fzval3 13094 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℤ → (1...(♯‘𝐴)) = (1..^((♯‘𝐴) + 1)))
4038, 39syl 17 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (1...(♯‘𝐴)) = (1..^((♯‘𝐴) + 1)))
4136, 40eqtrd 2853 . . . . . . . . 9 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → dom 𝑓 = (1..^((♯‘𝐴) + 1)))
4241olcd 870 . . . . . . . 8 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))
4335, 42jca 512 . . . . . . 7 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
4443ex 413 . . . . . 6 ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))))
4544eximdv 1909 . . . . 5 ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))))
4645imp 407 . . . 4 (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
4746adantl 482 . . 3 (((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) ∧ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
48 fz1f1o 15055 . . . 4 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
4948adantl 482 . . 3 ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
5031, 47, 49mpjaodan 952 . 2 ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
51 isfinite 9103 . . . . . . . . . 10 (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
5251notbii 321 . . . . . . . . 9 𝐴 ∈ Fin ↔ ¬ 𝐴 ≺ ω)
5352biimpi 217 . . . . . . . 8 𝐴 ∈ Fin → ¬ 𝐴 ≺ ω)
5453anim2i 616 . . . . . . 7 ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
55 bren2 8528 . . . . . . 7 (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
5654, 55sylibr 235 . . . . . 6 ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ω)
57 nnenom 13336 . . . . . . 7 ℕ ≈ ω
5857ensymi 8547 . . . . . 6 ω ≈ ℕ
59 entr 8549 . . . . . 6 ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ)
6056, 58, 59sylancl 586 . . . . 5 ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ℕ)
61 bren 8506 . . . . 5 (𝐴 ≈ ℕ ↔ ∃𝑔 𝑔:𝐴1-1-onto→ℕ)
6260, 61sylib 219 . . . 4 ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑔 𝑔:𝐴1-1-onto→ℕ)
63 f1oexbi 7622 . . . 4 (∃𝑔 𝑔:𝐴1-1-onto→ℕ ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
6462, 63sylib 219 . . 3 ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
65 f1odm 6612 . . . . . . 7 (𝑓:ℕ–1-1-onto𝐴 → dom 𝑓 = ℕ)
6665f1oeq2d 6604 . . . . . 6 (𝑓:ℕ–1-1-onto𝐴 → (𝑓:dom 𝑓1-1-onto𝐴𝑓:ℕ–1-1-onto𝐴))
6766ibir 269 . . . . 5 (𝑓:ℕ–1-1-onto𝐴𝑓:dom 𝑓1-1-onto𝐴)
6865orcd 869 . . . . 5 (𝑓:ℕ–1-1-onto𝐴 → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))
6967, 68jca 512 . . . 4 (𝑓:ℕ–1-1-onto𝐴 → (𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
7069eximi 1826 . . 3 (∃𝑓 𝑓:ℕ–1-1-onto𝐴 → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
7164, 70syl 17 . 2 ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
7250, 71pm2.61dan 809 1 (𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wex 1771  wcel 2105  c0 4288   class class class wbr 5057  dom cdm 5548  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  ωcom 7569  cen 8494  cdom 8495  csdm 8496  Fincfn 8497  0cc0 10525  1c1 10526   + caddc 10528  cn 11626  cz 11969  ...cfz 12880  ..^cfzo 13021  chash 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679
This theorem is referenced by: (None)
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