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Theorem fin1a2lem6 9428
Description: Lemma for fin1a2 9438. Establish that ω can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypotheses
Ref Expression
fin1a2lem.b 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
fin1a2lem.aa 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
Assertion
Ref Expression
fin1a2lem6 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)

Proof of Theorem fin1a2lem6
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.aa . . . 4 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
21fin1a2lem2 9424 . . 3 𝑆:On–1-1→On
3 fin1a2lem.b . . . . 5 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
43fin1a2lem4 9426 . . . 4 𝐸:ω–1-1→ω
5 f1f 6241 . . . 4 (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
6 frn 6193 . . . . 5 (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
7 omsson 7215 . . . . 5 ω ⊆ On
86, 7syl6ss 3762 . . . 4 (𝐸:ω⟶ω → ran 𝐸 ⊆ On)
94, 5, 8mp2b 10 . . 3 ran 𝐸 ⊆ On
10 f1ores 6292 . . 3 ((𝑆:On–1-1→On ∧ ran 𝐸 ⊆ On) → (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸))
112, 9, 10mp2an 664 . 2 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸)
129sseli 3746 . . . . . . . . 9 (𝑏 ∈ ran 𝐸𝑏 ∈ On)
131fin1a2lem1 9423 . . . . . . . . 9 (𝑏 ∈ On → (𝑆𝑏) = suc 𝑏)
1412, 13syl 17 . . . . . . . 8 (𝑏 ∈ ran 𝐸 → (𝑆𝑏) = suc 𝑏)
1514eqeq1d 2772 . . . . . . 7 (𝑏 ∈ ran 𝐸 → ((𝑆𝑏) = 𝑎 ↔ suc 𝑏 = 𝑎))
1615rexbiia 3187 . . . . . 6 (∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎 ↔ ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)
174, 5, 6mp2b 10 . . . . . . . . . . . 12 ran 𝐸 ⊆ ω
1817sseli 3746 . . . . . . . . . . 11 (𝑏 ∈ ran 𝐸𝑏 ∈ ω)
19 peano2 7232 . . . . . . . . . . 11 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
2018, 19syl 17 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 → suc 𝑏 ∈ ω)
213fin1a2lem5 9427 . . . . . . . . . . . 12 (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
2221biimpd 219 . . . . . . . . . . 11 (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸))
2318, 22mpcom 38 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸)
2420, 23jca 495 . . . . . . . . 9 (𝑏 ∈ ran 𝐸 → (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸))
25 eleq1 2837 . . . . . . . . . 10 (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ω ↔ 𝑎 ∈ ω))
26 eleq1 2837 . . . . . . . . . . 11 (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ran 𝐸𝑎 ∈ ran 𝐸))
2726notbid 307 . . . . . . . . . 10 (suc 𝑏 = 𝑎 → (¬ suc 𝑏 ∈ ran 𝐸 ↔ ¬ 𝑎 ∈ ran 𝐸))
2825, 27anbi12d 608 . . . . . . . . 9 (suc 𝑏 = 𝑎 → ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)))
2924, 28syl5ibcom 235 . . . . . . . 8 (𝑏 ∈ ran 𝐸 → (suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)))
3029rexlimiv 3174 . . . . . . 7 (∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
31 peano1 7231 . . . . . . . . . . . . . 14 ∅ ∈ ω
323fin1a2lem3 9425 . . . . . . . . . . . . . 14 (∅ ∈ ω → (𝐸‘∅) = (2𝑜 ·𝑜 ∅))
3331, 32ax-mp 5 . . . . . . . . . . . . 13 (𝐸‘∅) = (2𝑜 ·𝑜 ∅)
34 om0x 7752 . . . . . . . . . . . . 13 (2𝑜 ·𝑜 ∅) = ∅
3533, 34eqtri 2792 . . . . . . . . . . . 12 (𝐸‘∅) = ∅
36 f1fun 6243 . . . . . . . . . . . . . 14 (𝐸:ω–1-1→ω → Fun 𝐸)
374, 36ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐸
38 f1dm 6245 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1→ω → dom 𝐸 = ω)
394, 38ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐸 = ω
4031, 39eleqtrri 2848 . . . . . . . . . . . . 13 ∅ ∈ dom 𝐸
41 fvelrn 6495 . . . . . . . . . . . . 13 ((Fun 𝐸 ∧ ∅ ∈ dom 𝐸) → (𝐸‘∅) ∈ ran 𝐸)
4237, 40, 41mp2an 664 . . . . . . . . . . . 12 (𝐸‘∅) ∈ ran 𝐸
4335, 42eqeltrri 2846 . . . . . . . . . . 11 ∅ ∈ ran 𝐸
44 eleq1 2837 . . . . . . . . . . 11 (𝑎 = ∅ → (𝑎 ∈ ran 𝐸 ↔ ∅ ∈ ran 𝐸))
4543, 44mpbiri 248 . . . . . . . . . 10 (𝑎 = ∅ → 𝑎 ∈ ran 𝐸)
4645necon3bi 2968 . . . . . . . . 9 𝑎 ∈ ran 𝐸𝑎 ≠ ∅)
47 nnsuc 7228 . . . . . . . . 9 ((𝑎 ∈ ω ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
4846, 47sylan2 572 . . . . . . . 8 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
49 eleq1 2837 . . . . . . . . . . . . . . . 16 (𝑎 = suc 𝑏 → (𝑎 ∈ ω ↔ suc 𝑏 ∈ ω))
50 eleq1 2837 . . . . . . . . . . . . . . . . 17 (𝑎 = suc 𝑏 → (𝑎 ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸))
5150notbid 307 . . . . . . . . . . . . . . . 16 (𝑎 = suc 𝑏 → (¬ 𝑎 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
5249, 51anbi12d 608 . . . . . . . . . . . . . . 15 (𝑎 = suc 𝑏 → ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ↔ (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸)))
5352anbi1d 607 . . . . . . . . . . . . . 14 (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) ↔ ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω)))
54 simplr 744 . . . . . . . . . . . . . . 15 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → ¬ suc 𝑏 ∈ ran 𝐸)
5521adantl 467 . . . . . . . . . . . . . . 15 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
5654, 55mpbird 247 . . . . . . . . . . . . . 14 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸)
5753, 56syl6bi 243 . . . . . . . . . . . . 13 (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸))
5857com12 32 . . . . . . . . . . . 12 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → (𝑎 = suc 𝑏𝑏 ∈ ran 𝐸))
5958impr 442 . . . . . . . . . . 11 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑏 ∈ ran 𝐸)
60 simprr 748 . . . . . . . . . . . 12 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑎 = suc 𝑏)
6160eqcomd 2776 . . . . . . . . . . 11 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → suc 𝑏 = 𝑎)
6259, 61jca 495 . . . . . . . . . 10 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → (𝑏 ∈ ran 𝐸 ∧ suc 𝑏 = 𝑎))
6362ex 397 . . . . . . . . 9 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ((𝑏 ∈ ω ∧ 𝑎 = suc 𝑏) → (𝑏 ∈ ran 𝐸 ∧ suc 𝑏 = 𝑎)))
6463reximdv2 3161 . . . . . . . 8 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎))
6548, 64mpd 15 . . . . . . 7 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)
6630, 65impbii 199 . . . . . 6 (∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
6716, 66bitri 264 . . . . 5 (∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
68 f1fn 6242 . . . . . . 7 (𝑆:On–1-1→On → 𝑆 Fn On)
692, 68ax-mp 5 . . . . . 6 𝑆 Fn On
70 fvelimab 6395 . . . . . 6 ((𝑆 Fn On ∧ ran 𝐸 ⊆ On) → (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎))
7169, 9, 70mp2an 664 . . . . 5 (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎)
72 eldif 3731 . . . . 5 (𝑎 ∈ (ω ∖ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
7367, 71, 723bitr4i 292 . . . 4 (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ 𝑎 ∈ (ω ∖ ran 𝐸))
7473eqriv 2767 . . 3 (𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸)
75 f1oeq3 6270 . . 3 ((𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸) → ((𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸) ↔ (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)))
7674, 75ax-mp 5 . 2 ((𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸) ↔ (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸))
7711, 76mpbi 220 1 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382   = wceq 1630  wcel 2144  wne 2942  wrex 3061  cdif 3718  wss 3721  c0 4061  cmpt 4861  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  Oncon0 5866  suc csuc 5868  Fun wfun 6025   Fn wfn 6026  wf 6027  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  ωcom 7211  2𝑜c2o 7706   ·𝑜 comu 7710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-omul 7717
This theorem is referenced by:  fin1a2lem7  9429
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