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Theorem fin1a2lem6 9815
Description: Lemma for fin1a2 9825. 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 9811 . . 3 𝑆:On–1-1→On
3 fin1a2lem.b . . . . 5 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
43fin1a2lem4 9813 . . . 4 𝐸:ω–1-1→ω
5 f1f 6568 . . . 4 (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
6 frn 6513 . . . . 5 (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
7 omsson 7573 . . . . 5 ω ⊆ On
86, 7sstrdi 3976 . . . 4 (𝐸:ω⟶ω → ran 𝐸 ⊆ On)
94, 5, 8mp2b 10 . . 3 ran 𝐸 ⊆ On
10 f1ores 6622 . . 3 ((𝑆:On–1-1→On ∧ ran 𝐸 ⊆ On) → (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸))
112, 9, 10mp2an 688 . 2 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸)
129sseli 3960 . . . . . . . . 9 (𝑏 ∈ ran 𝐸𝑏 ∈ On)
131fin1a2lem1 9810 . . . . . . . . 9 (𝑏 ∈ On → (𝑆𝑏) = suc 𝑏)
1412, 13syl 17 . . . . . . . 8 (𝑏 ∈ ran 𝐸 → (𝑆𝑏) = suc 𝑏)
1514eqeq1d 2820 . . . . . . 7 (𝑏 ∈ ran 𝐸 → ((𝑆𝑏) = 𝑎 ↔ suc 𝑏 = 𝑎))
1615rexbiia 3243 . . . . . 6 (∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎 ↔ ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)
174, 5, 6mp2b 10 . . . . . . . . . . . 12 ran 𝐸 ⊆ ω
1817sseli 3960 . . . . . . . . . . 11 (𝑏 ∈ ran 𝐸𝑏 ∈ ω)
19 peano2 7591 . . . . . . . . . . 11 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
2018, 19syl 17 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 → suc 𝑏 ∈ ω)
213fin1a2lem5 9814 . . . . . . . . . . . 12 (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
2221biimpd 230 . . . . . . . . . . 11 (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸))
2318, 22mpcom 38 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸)
2420, 23jca 512 . . . . . . . . 9 (𝑏 ∈ ran 𝐸 → (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸))
25 eleq1 2897 . . . . . . . . . 10 (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ω ↔ 𝑎 ∈ ω))
26 eleq1 2897 . . . . . . . . . . 11 (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ran 𝐸𝑎 ∈ ran 𝐸))
2726notbid 319 . . . . . . . . . 10 (suc 𝑏 = 𝑎 → (¬ suc 𝑏 ∈ ran 𝐸 ↔ ¬ 𝑎 ∈ ran 𝐸))
2825, 27anbi12d 630 . . . . . . . . 9 (suc 𝑏 = 𝑎 → ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)))
2924, 28syl5ibcom 246 . . . . . . . 8 (𝑏 ∈ ran 𝐸 → (suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)))
3029rexlimiv 3277 . . . . . . 7 (∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
31 peano1 7590 . . . . . . . . . . . . . 14 ∅ ∈ ω
323fin1a2lem3 9812 . . . . . . . . . . . . . 14 (∅ ∈ ω → (𝐸‘∅) = (2o ·o ∅))
3331, 32ax-mp 5 . . . . . . . . . . . . 13 (𝐸‘∅) = (2o ·o ∅)
34 2on 8100 . . . . . . . . . . . . . 14 2o ∈ On
35 om0 8131 . . . . . . . . . . . . . 14 (2o ∈ On → (2o ·o ∅) = ∅)
3634, 35ax-mp 5 . . . . . . . . . . . . 13 (2o ·o ∅) = ∅
3733, 36eqtri 2841 . . . . . . . . . . . 12 (𝐸‘∅) = ∅
38 f1fun 6570 . . . . . . . . . . . . . 14 (𝐸:ω–1-1→ω → Fun 𝐸)
394, 38ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐸
40 f1dm 6572 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1→ω → dom 𝐸 = ω)
414, 40ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐸 = ω
4231, 41eleqtrri 2909 . . . . . . . . . . . . 13 ∅ ∈ dom 𝐸
43 fvelrn 6836 . . . . . . . . . . . . 13 ((Fun 𝐸 ∧ ∅ ∈ dom 𝐸) → (𝐸‘∅) ∈ ran 𝐸)
4439, 42, 43mp2an 688 . . . . . . . . . . . 12 (𝐸‘∅) ∈ ran 𝐸
4537, 44eqeltrri 2907 . . . . . . . . . . 11 ∅ ∈ ran 𝐸
46 eleq1 2897 . . . . . . . . . . 11 (𝑎 = ∅ → (𝑎 ∈ ran 𝐸 ↔ ∅ ∈ ran 𝐸))
4745, 46mpbiri 259 . . . . . . . . . 10 (𝑎 = ∅ → 𝑎 ∈ ran 𝐸)
4847necon3bi 3039 . . . . . . . . 9 𝑎 ∈ ran 𝐸𝑎 ≠ ∅)
49 nnsuc 7586 . . . . . . . . 9 ((𝑎 ∈ ω ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
5048, 49sylan2 592 . . . . . . . 8 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
51 eleq1 2897 . . . . . . . . . . . . 13 (𝑎 = suc 𝑏 → (𝑎 ∈ ω ↔ suc 𝑏 ∈ ω))
52 eleq1 2897 . . . . . . . . . . . . . 14 (𝑎 = suc 𝑏 → (𝑎 ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸))
5352notbid 319 . . . . . . . . . . . . 13 (𝑎 = suc 𝑏 → (¬ 𝑎 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
5451, 53anbi12d 630 . . . . . . . . . . . 12 (𝑎 = suc 𝑏 → ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ↔ (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸)))
5554anbi1d 629 . . . . . . . . . . 11 (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) ↔ ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω)))
56 simplr 765 . . . . . . . . . . . 12 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → ¬ suc 𝑏 ∈ ran 𝐸)
5721adantl 482 . . . . . . . . . . . 12 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
5856, 57mpbird 258 . . . . . . . . . . 11 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸)
5955, 58syl6bi 254 . . . . . . . . . 10 (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸))
6059com12 32 . . . . . . . . 9 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → (𝑎 = suc 𝑏𝑏 ∈ ran 𝐸))
6160impr 455 . . . . . . . 8 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑏 ∈ ran 𝐸)
62 simprr 769 . . . . . . . . 9 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑎 = suc 𝑏)
6362eqcomd 2824 . . . . . . . 8 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → suc 𝑏 = 𝑎)
6450, 61, 63reximssdv 3273 . . . . . . 7 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)
6530, 64impbii 210 . . . . . 6 (∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
6616, 65bitri 276 . . . . 5 (∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
67 f1fn 6569 . . . . . . 7 (𝑆:On–1-1→On → 𝑆 Fn On)
682, 67ax-mp 5 . . . . . 6 𝑆 Fn On
69 fvelimab 6730 . . . . . 6 ((𝑆 Fn On ∧ ran 𝐸 ⊆ On) → (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎))
7068, 9, 69mp2an 688 . . . . 5 (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎)
71 eldif 3943 . . . . 5 (𝑎 ∈ (ω ∖ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
7266, 70, 713bitr4i 304 . . . 4 (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ 𝑎 ∈ (ω ∖ ran 𝐸))
7372eqriv 2815 . . 3 (𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸)
74 f1oeq3 6599 . . 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)))
7573, 74ax-mp 5 . 2 ((𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸) ↔ (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸))
7611, 75mpbi 231 1 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wrex 3136  cdif 3930  wss 3933  c0 4288  cmpt 5137  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Oncon0 6184  suc csuc 6186  Fun wfun 6342   Fn wfn 6343  wf 6344  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  ωcom 7569  2oc2o 8085   ·o comu 8089
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-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
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-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-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-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096
This theorem is referenced by:  fin1a2lem7  9816
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