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Theorem unbenlem 15450
Description: Lemma for unben 15451. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)
Hypothesis
Ref Expression
unbenlem.1 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)
Assertion
Ref Expression
unbenlem ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → 𝐴 ≈ ω)
Distinct variable groups:   𝑚,𝑛,𝐴   𝑚,𝐺,𝑛
Allowed substitution hints:   𝐴(𝑥)   𝐺(𝑥)

Proof of Theorem unbenlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nnex 10903 . . . . 5 ℕ ∈ V
21ssex 4730 . . . 4 (𝐴 ⊆ ℕ → 𝐴 ∈ V)
3 1z 11284 . . . . . . . 8 1 ∈ ℤ
4 unbenlem.1 . . . . . . . 8 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)
53, 4om2uzf1oi 12614 . . . . . . 7 𝐺:ω–1-1-onto→(ℤ‘1)
6 nnuz 11599 . . . . . . . 8 ℕ = (ℤ‘1)
7 f1oeq3 6042 . . . . . . . 8 (ℕ = (ℤ‘1) → (𝐺:ω–1-1-onto→ℕ ↔ 𝐺:ω–1-1-onto→(ℤ‘1)))
86, 7ax-mp 5 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ ↔ 𝐺:ω–1-1-onto→(ℤ‘1))
95, 8mpbir 220 . . . . . 6 𝐺:ω–1-1-onto→ℕ
10 f1ocnv 6062 . . . . . 6 (𝐺:ω–1-1-onto→ℕ → 𝐺:ℕ–1-1-onto→ω)
11 f1of1 6049 . . . . . 6 (𝐺:ℕ–1-1-onto→ω → 𝐺:ℕ–1-1→ω)
129, 10, 11mp2b 10 . . . . 5 𝐺:ℕ–1-1→ω
13 f1ores 6064 . . . . 5 ((𝐺:ℕ–1-1→ω ∧ 𝐴 ⊆ ℕ) → (𝐺𝐴):𝐴1-1-onto→(𝐺𝐴))
1412, 13mpan 702 . . . 4 (𝐴 ⊆ ℕ → (𝐺𝐴):𝐴1-1-onto→(𝐺𝐴))
15 f1oeng 7860 . . . 4 ((𝐴 ∈ V ∧ (𝐺𝐴):𝐴1-1-onto→(𝐺𝐴)) → 𝐴 ≈ (𝐺𝐴))
162, 14, 15syl2anc 691 . . 3 (𝐴 ⊆ ℕ → 𝐴 ≈ (𝐺𝐴))
1716adantr 480 . 2 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → 𝐴 ≈ (𝐺𝐴))
18 imassrn 5396 . . . 4 (𝐺𝐴) ⊆ ran 𝐺
19 dfdm4 5238 . . . . 5 dom 𝐺 = ran 𝐺
20 f1of 6050 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ → 𝐺:ω⟶ℕ)
219, 20ax-mp 5 . . . . . 6 𝐺:ω⟶ℕ
2221fdmi 5965 . . . . 5 dom 𝐺 = ω
2319, 22eqtr3i 2634 . . . 4 ran 𝐺 = ω
2418, 23sseqtri 3600 . . 3 (𝐺𝐴) ⊆ ω
253, 4om2uzuzi 12610 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝐺𝑦) ∈ (ℤ‘1))
2625, 6syl6eleqr 2699 . . . . . . . . . 10 (𝑦 ∈ ω → (𝐺𝑦) ∈ ℕ)
27 breq1 4586 . . . . . . . . . . . 12 (𝑚 = (𝐺𝑦) → (𝑚 < 𝑛 ↔ (𝐺𝑦) < 𝑛))
2827rexbidv 3034 . . . . . . . . . . 11 (𝑚 = (𝐺𝑦) → (∃𝑛𝐴 𝑚 < 𝑛 ↔ ∃𝑛𝐴 (𝐺𝑦) < 𝑛))
2928rspcv 3278 . . . . . . . . . 10 ((𝐺𝑦) ∈ ℕ → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → ∃𝑛𝐴 (𝐺𝑦) < 𝑛))
3026, 29syl 17 . . . . . . . . 9 (𝑦 ∈ ω → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → ∃𝑛𝐴 (𝐺𝑦) < 𝑛))
3130adantr 480 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → ∃𝑛𝐴 (𝐺𝑦) < 𝑛))
32 f1ocnv 6062 . . . . . . . . . . . . . . . . 17 ((𝐺𝐴):𝐴1-1-onto→(𝐺𝐴) → (𝐺𝐴):(𝐺𝐴)–1-1-onto𝐴)
3314, 32syl 17 . . . . . . . . . . . . . . . 16 (𝐴 ⊆ ℕ → (𝐺𝐴):(𝐺𝐴)–1-1-onto𝐴)
34 f1ofun 6052 . . . . . . . . . . . . . . . . . 18 (𝐺:ω–1-1-onto→ℕ → Fun 𝐺)
359, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 Fun 𝐺
36 funcnvres2 5883 . . . . . . . . . . . . . . . . 17 (Fun 𝐺(𝐺𝐴) = (𝐺 ↾ (𝐺𝐴)))
37 f1oeq1 6040 . . . . . . . . . . . . . . . . 17 ((𝐺𝐴) = (𝐺 ↾ (𝐺𝐴)) → ((𝐺𝐴):(𝐺𝐴)–1-1-onto𝐴 ↔ (𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴))
3835, 36, 37mp2b 10 . . . . . . . . . . . . . . . 16 ((𝐺𝐴):(𝐺𝐴)–1-1-onto𝐴 ↔ (𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴)
3933, 38sylib 207 . . . . . . . . . . . . . . 15 (𝐴 ⊆ ℕ → (𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴)
40 f1ofo 6057 . . . . . . . . . . . . . . . . . 18 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → (𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–onto𝐴)
41 forn 6031 . . . . . . . . . . . . . . . . . 18 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–onto𝐴 → ran (𝐺 ↾ (𝐺𝐴)) = 𝐴)
4240, 41syl 17 . . . . . . . . . . . . . . . . 17 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → ran (𝐺 ↾ (𝐺𝐴)) = 𝐴)
4342eleq2d 2673 . . . . . . . . . . . . . . . 16 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → (𝑛 ∈ ran (𝐺 ↾ (𝐺𝐴)) ↔ 𝑛𝐴))
44 f1ofn 6051 . . . . . . . . . . . . . . . . 17 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → (𝐺 ↾ (𝐺𝐴)) Fn (𝐺𝐴))
45 fvelrnb 6153 . . . . . . . . . . . . . . . . 17 ((𝐺 ↾ (𝐺𝐴)) Fn (𝐺𝐴) → (𝑛 ∈ ran (𝐺 ↾ (𝐺𝐴)) ↔ ∃𝑚 ∈ (𝐺𝐴)((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛))
4644, 45syl 17 . . . . . . . . . . . . . . . 16 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → (𝑛 ∈ ran (𝐺 ↾ (𝐺𝐴)) ↔ ∃𝑚 ∈ (𝐺𝐴)((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛))
4743, 46bitr3d 269 . . . . . . . . . . . . . . 15 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → (𝑛𝐴 ↔ ∃𝑚 ∈ (𝐺𝐴)((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛))
4839, 47syl 17 . . . . . . . . . . . . . 14 (𝐴 ⊆ ℕ → (𝑛𝐴 ↔ ∃𝑚 ∈ (𝐺𝐴)((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛))
4948biimpa 500 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℕ ∧ 𝑛𝐴) → ∃𝑚 ∈ (𝐺𝐴)((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛)
50 fvres 6117 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (𝐺𝐴) → ((𝐺 ↾ (𝐺𝐴))‘𝑚) = (𝐺𝑚))
5150eqeq1d 2612 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (𝐺𝐴) → (((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛 ↔ (𝐺𝑚) = 𝑛))
5251biimpa 500 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ (𝐺𝐴) ∧ ((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛) → (𝐺𝑚) = 𝑛)
5352adantll 746 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) ∧ ((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛) → (𝐺𝑚) = 𝑛)
5424sseli 3564 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (𝐺𝐴) → 𝑚 ∈ ω)
553, 4om2uzlt2i 12612 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ω ∧ 𝑚 ∈ ω) → (𝑦𝑚 ↔ (𝐺𝑦) < (𝐺𝑚)))
5654, 55sylan2 490 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) → (𝑦𝑚 ↔ (𝐺𝑦) < (𝐺𝑚)))
57 breq2 4587 . . . . . . . . . . . . . . . . . . 19 ((𝐺𝑚) = 𝑛 → ((𝐺𝑦) < (𝐺𝑚) ↔ (𝐺𝑦) < 𝑛))
5856, 57sylan9bb 732 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) ∧ (𝐺𝑚) = 𝑛) → (𝑦𝑚 ↔ (𝐺𝑦) < 𝑛))
5953, 58syldan 486 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) ∧ ((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛) → (𝑦𝑚 ↔ (𝐺𝑦) < 𝑛))
6059biimparc 503 . . . . . . . . . . . . . . . 16 (((𝐺𝑦) < 𝑛 ∧ ((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) ∧ ((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛)) → 𝑦𝑚)
6160exp44 639 . . . . . . . . . . . . . . 15 ((𝐺𝑦) < 𝑛 → (𝑦 ∈ ω → (𝑚 ∈ (𝐺𝐴) → (((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛𝑦𝑚))))
6261imp31 447 . . . . . . . . . . . . . 14 ((((𝐺𝑦) < 𝑛𝑦 ∈ ω) ∧ 𝑚 ∈ (𝐺𝐴)) → (((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛𝑦𝑚))
6362reximdva 3000 . . . . . . . . . . . . 13 (((𝐺𝑦) < 𝑛𝑦 ∈ ω) → (∃𝑚 ∈ (𝐺𝐴)((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))
6449, 63syl5 33 . . . . . . . . . . . 12 (((𝐺𝑦) < 𝑛𝑦 ∈ ω) → ((𝐴 ⊆ ℕ ∧ 𝑛𝐴) → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))
6564exp4b 630 . . . . . . . . . . 11 ((𝐺𝑦) < 𝑛 → (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (𝑛𝐴 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))))
6665com4l 90 . . . . . . . . . 10 (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (𝑛𝐴 → ((𝐺𝑦) < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))))
6766imp 444 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (𝑛𝐴 → ((𝐺𝑦) < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚)))
6867rexlimdv 3012 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∃𝑛𝐴 (𝐺𝑦) < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))
6931, 68syld 46 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))
7069ex 449 . . . . . 6 (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚)))
7170com3l 87 . . . . 5 (𝐴 ⊆ ℕ → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → (𝑦 ∈ ω → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚)))
7271imp 444 . . . 4 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → (𝑦 ∈ ω → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))
7372ralrimiv 2948 . . 3 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → ∀𝑦 ∈ ω ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚)
74 unbnn3 8439 . . 3 (((𝐺𝐴) ⊆ ω ∧ ∀𝑦 ∈ ω ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚) → (𝐺𝐴) ≈ ω)
7524, 73, 74sylancr 694 . 2 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → (𝐺𝐴) ≈ ω)
76 entr 7894 . 2 ((𝐴 ≈ (𝐺𝐴) ∧ (𝐺𝐴) ≈ ω) → 𝐴 ≈ ω)
7717, 75, 76syl2anc 691 1 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → 𝐴 ≈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  wss 3540   class class class wbr 4583  cmpt 4643  ccnv 5037  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Fun wfun 5798   Fn wfn 5799  wf 5800  1-1wf1 5801  ontowfo 5802  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  ωcom 6957  reccrdg 7392  cen 7838  1c1 9816   + caddc 9818   < clt 9953  cn 10897  cuz 11563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564
This theorem is referenced by:  unben  15451
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