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Theorem unbenlem 15334
Description: Lemma for unben 15335. (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 10781 . . . . 5 ℕ ∈ V
21ssex 4629 . . . 4 (𝐴 ⊆ ℕ → 𝐴 ∈ V)
3 1z 11148 . . . . . . . 8 1 ∈ ℤ
4 unbenlem.1 . . . . . . . 8 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)
53, 4om2uzf1oi 12482 . . . . . . 7 𝐺:ω–1-1-onto→(ℤ‘1)
6 nnuz 11463 . . . . . . . 8 ℕ = (ℤ‘1)
7 f1oeq3 5926 . . . . . . . 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 219 . . . . . 6 𝐺:ω–1-1-onto→ℕ
10 f1ocnv 5946 . . . . . 6 (𝐺:ω–1-1-onto→ℕ → 𝐺:ℕ–1-1-onto→ω)
11 f1of1 5933 . . . . . 6 (𝐺:ℕ–1-1-onto→ω → 𝐺:ℕ–1-1→ω)
129, 10, 11mp2b 10 . . . . 5 𝐺:ℕ–1-1→ω
13 f1ores 5948 . . . . 5 ((𝐺:ℕ–1-1→ω ∧ 𝐴 ⊆ ℕ) → (𝐺𝐴):𝐴1-1-onto→(𝐺𝐴))
1412, 13mpan 701 . . . 4 (𝐴 ⊆ ℕ → (𝐺𝐴):𝐴1-1-onto→(𝐺𝐴))
15 f1oeng 7736 . . . 4 ((𝐴 ∈ V ∧ (𝐺𝐴):𝐴1-1-onto→(𝐺𝐴)) → 𝐴 ≈ (𝐺𝐴))
162, 14, 15syl2anc 690 . . 3 (𝐴 ⊆ ℕ → 𝐴 ≈ (𝐺𝐴))
1716adantr 479 . 2 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → 𝐴 ≈ (𝐺𝐴))
18 imassrn 5287 . . . 4 (𝐺𝐴) ⊆ ran 𝐺
19 dfdm4 5129 . . . . 5 dom 𝐺 = ran 𝐺
20 f1of 5934 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ → 𝐺:ω⟶ℕ)
219, 20ax-mp 5 . . . . . 6 𝐺:ω⟶ℕ
2221fdmi 5850 . . . . 5 dom 𝐺 = ω
2319, 22eqtr3i 2538 . . . 4 ran 𝐺 = ω
2418, 23sseqtri 3504 . . 3 (𝐺𝐴) ⊆ ω
253, 4om2uzuzi 12478 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝐺𝑦) ∈ (ℤ‘1))
2625, 6syl6eleqr 2603 . . . . . . . . . 10 (𝑦 ∈ ω → (𝐺𝑦) ∈ ℕ)
27 breq1 4484 . . . . . . . . . . . 12 (𝑚 = (𝐺𝑦) → (𝑚 < 𝑛 ↔ (𝐺𝑦) < 𝑛))
2827rexbidv 2938 . . . . . . . . . . 11 (𝑚 = (𝐺𝑦) → (∃𝑛𝐴 𝑚 < 𝑛 ↔ ∃𝑛𝐴 (𝐺𝑦) < 𝑛))
2928rspcv 3182 . . . . . . . . . 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 479 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → ∃𝑛𝐴 (𝐺𝑦) < 𝑛))
32 f1ocnv 5946 . . . . . . . . . . . . . . . . 17 ((𝐺𝐴):𝐴1-1-onto→(𝐺𝐴) → (𝐺𝐴):(𝐺𝐴)–1-1-onto𝐴)
3314, 32syl 17 . . . . . . . . . . . . . . . 16 (𝐴 ⊆ ℕ → (𝐺𝐴):(𝐺𝐴)–1-1-onto𝐴)
34 f1ofun 5936 . . . . . . . . . . . . . . . . . 18 (𝐺:ω–1-1-onto→ℕ → Fun 𝐺)
359, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 Fun 𝐺
36 funcnvres2 5768 . . . . . . . . . . . . . . . . 17 (Fun 𝐺(𝐺𝐴) = (𝐺 ↾ (𝐺𝐴)))
37 f1oeq1 5924 . . . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . . 15 (𝐴 ⊆ ℕ → (𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴)
40 f1ofo 5941 . . . . . . . . . . . . . . . . . 18 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → (𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–onto𝐴)
41 forn 5915 . . . . . . . . . . . . . . . . . 18 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–onto𝐴 → ran (𝐺 ↾ (𝐺𝐴)) = 𝐴)
4240, 41syl 17 . . . . . . . . . . . . . . . . 17 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → ran (𝐺 ↾ (𝐺𝐴)) = 𝐴)
4342eleq2d 2577 . . . . . . . . . . . . . . . 16 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → (𝑛 ∈ ran (𝐺 ↾ (𝐺𝐴)) ↔ 𝑛𝐴))
44 f1ofn 5935 . . . . . . . . . . . . . . . . 17 ((𝐺 ↾ (𝐺𝐴)):(𝐺𝐴)–1-1-onto𝐴 → (𝐺 ↾ (𝐺𝐴)) Fn (𝐺𝐴))
45 fvelrnb 6037 . . . . . . . . . . . . . . . . 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 268 . . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℕ ∧ 𝑛𝐴) → ∃𝑚 ∈ (𝐺𝐴)((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛)
50 fvres 6001 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (𝐺𝐴) → ((𝐺 ↾ (𝐺𝐴))‘𝑚) = (𝐺𝑚))
5150eqeq1d 2516 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (𝐺𝐴) → (((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛 ↔ (𝐺𝑚) = 𝑛))
5251biimpa 499 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ (𝐺𝐴) ∧ ((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛) → (𝐺𝑚) = 𝑛)
5352adantll 745 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) ∧ ((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛) → (𝐺𝑚) = 𝑛)
5424sseli 3468 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (𝐺𝐴) → 𝑚 ∈ ω)
553, 4om2uzlt2i 12480 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ω ∧ 𝑚 ∈ ω) → (𝑦𝑚 ↔ (𝐺𝑦) < (𝐺𝑚)))
5654, 55sylan2 489 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) → (𝑦𝑚 ↔ (𝐺𝑦) < (𝐺𝑚)))
57 breq2 4485 . . . . . . . . . . . . . . . . . . 19 ((𝐺𝑚) = 𝑛 → ((𝐺𝑦) < (𝐺𝑚) ↔ (𝐺𝑦) < 𝑛))
5856, 57sylan9bb 731 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) ∧ (𝐺𝑚) = 𝑛) → (𝑦𝑚 ↔ (𝐺𝑦) < 𝑛))
5953, 58syldan 485 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) ∧ ((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛) → (𝑦𝑚 ↔ (𝐺𝑦) < 𝑛))
6059biimparc 502 . . . . . . . . . . . . . . . 16 (((𝐺𝑦) < 𝑛 ∧ ((𝑦 ∈ ω ∧ 𝑚 ∈ (𝐺𝐴)) ∧ ((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛)) → 𝑦𝑚)
6160exp44 638 . . . . . . . . . . . . . . 15 ((𝐺𝑦) < 𝑛 → (𝑦 ∈ ω → (𝑚 ∈ (𝐺𝐴) → (((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛𝑦𝑚))))
6261imp31 446 . . . . . . . . . . . . . 14 ((((𝐺𝑦) < 𝑛𝑦 ∈ ω) ∧ 𝑚 ∈ (𝐺𝐴)) → (((𝐺 ↾ (𝐺𝐴))‘𝑚) = 𝑛𝑦𝑚))
6362reximdva 2904 . . . . . . . . . . . . 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 629 . . . . . . . . . . 11 ((𝐺𝑦) < 𝑛 → (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (𝑛𝐴 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))))
6665com4l 89 . . . . . . . . . 10 (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (𝑛𝐴 → ((𝐺𝑦) < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))))
6766imp 443 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (𝑛𝐴 → ((𝐺𝑦) < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚)))
6867rexlimdv 2916 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∃𝑛𝐴 (𝐺𝑦) < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))
6931, 68syld 45 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐴 ⊆ ℕ) → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))
7069ex 448 . . . . . 6 (𝑦 ∈ ω → (𝐴 ⊆ ℕ → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚)))
7170com3l 86 . . . . 5 (𝐴 ⊆ ℕ → (∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛 → (𝑦 ∈ ω → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚)))
7271imp 443 . . . 4 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → (𝑦 ∈ ω → ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚))
7372ralrimiv 2852 . . 3 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → ∀𝑦 ∈ ω ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚)
74 unbnn3 8315 . . 3 (((𝐺𝐴) ⊆ ω ∧ ∀𝑦 ∈ ω ∃𝑚 ∈ (𝐺𝐴)𝑦𝑚) → (𝐺𝐴) ≈ ω)
7524, 73, 74sylancr 693 . 2 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → (𝐺𝐴) ≈ ω)
76 entr 7770 . 2 ((𝐴 ≈ (𝐺𝐴) ∧ (𝐺𝐴) ≈ ω) → 𝐴 ≈ ω)
7717, 75, 76syl2anc 690 1 ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → 𝐴 ≈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  wral 2800  wrex 2801  Vcvv 3077  wss 3444   class class class wbr 4481  cmpt 4541  ccnv 4931  dom cdm 4932  ran crn 4933  cres 4934  cima 4935  Fun wfun 5683   Fn wfn 5684  wf 5685  1-1wf1 5686  ontowfo 5687  1-1-ontowf1o 5688  cfv 5689  (class class class)co 6426  ωcom 6833  reccrdg 7268  cen 7714  1c1 9692   + caddc 9694   < clt 9829  cn 10775  cuz 11427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-inf2 8297  ax-cnex 9747  ax-resscn 9748  ax-1cn 9749  ax-icn 9750  ax-addcl 9751  ax-addrcl 9752  ax-mulcl 9753  ax-mulrcl 9754  ax-mulcom 9755  ax-addass 9756  ax-mulass 9757  ax-distr 9758  ax-i2m1 9759  ax-1ne0 9760  ax-1rid 9761  ax-rnegex 9762  ax-rrecex 9763  ax-cnre 9764  ax-pre-lttri 9765  ax-pre-lttrn 9766  ax-pre-ltadd 9767  ax-pre-mulgt0 9768
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-om 6834  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-er 7505  df-en 7718  df-dom 7719  df-sdom 7720  df-pnf 9831  df-mnf 9832  df-xr 9833  df-ltxr 9834  df-le 9835  df-sub 10019  df-neg 10020  df-nn 10776  df-n0 11048  df-z 11119  df-uz 11428
This theorem is referenced by:  unben  15335
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