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Theorem fin1a2lem6 10442
Description: Lemma for fin1a2 10452. 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 10438 . . 3 𝑆:On–1-1→On
3 fin1a2lem.b . . . . 5 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
43fin1a2lem4 10440 . . . 4 𝐸:ω–1-1→ω
5 f1f 6804 . . . 4 (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
6 frn 6743 . . . . 5 (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
7 omsson 7890 . . . . 5 ω ⊆ On
86, 7sstrdi 4007 . . . 4 (𝐸:ω⟶ω → ran 𝐸 ⊆ On)
94, 5, 8mp2b 10 . . 3 ran 𝐸 ⊆ On
10 f1ores 6862 . . 3 ((𝑆:On–1-1→On ∧ ran 𝐸 ⊆ On) → (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸))
112, 9, 10mp2an 692 . 2 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸)
129sseli 3990 . . . . . . . . 9 (𝑏 ∈ ran 𝐸𝑏 ∈ On)
131fin1a2lem1 10437 . . . . . . . . 9 (𝑏 ∈ On → (𝑆𝑏) = suc 𝑏)
1412, 13syl 17 . . . . . . . 8 (𝑏 ∈ ran 𝐸 → (𝑆𝑏) = suc 𝑏)
1514eqeq1d 2736 . . . . . . 7 (𝑏 ∈ ran 𝐸 → ((𝑆𝑏) = 𝑎 ↔ suc 𝑏 = 𝑎))
1615rexbiia 3089 . . . . . 6 (∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎 ↔ ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)
174, 5, 6mp2b 10 . . . . . . . . . . . 12 ran 𝐸 ⊆ ω
1817sseli 3990 . . . . . . . . . . 11 (𝑏 ∈ ran 𝐸𝑏 ∈ ω)
19 peano2 7912 . . . . . . . . . . 11 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
2018, 19syl 17 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 → suc 𝑏 ∈ ω)
213fin1a2lem5 10441 . . . . . . . . . . . 12 (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
2221biimpd 229 . . . . . . . . . . 11 (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸))
2318, 22mpcom 38 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸)
2420, 23jca 511 . . . . . . . . 9 (𝑏 ∈ ran 𝐸 → (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸))
25 eleq1 2826 . . . . . . . . . 10 (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ω ↔ 𝑎 ∈ ω))
26 eleq1 2826 . . . . . . . . . . 11 (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ran 𝐸𝑎 ∈ ran 𝐸))
2726notbid 318 . . . . . . . . . 10 (suc 𝑏 = 𝑎 → (¬ suc 𝑏 ∈ ran 𝐸 ↔ ¬ 𝑎 ∈ ran 𝐸))
2825, 27anbi12d 632 . . . . . . . . 9 (suc 𝑏 = 𝑎 → ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)))
2924, 28syl5ibcom 245 . . . . . . . 8 (𝑏 ∈ ran 𝐸 → (suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)))
3029rexlimiv 3145 . . . . . . 7 (∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
31 peano1 7910 . . . . . . . . . . . . . 14 ∅ ∈ ω
323fin1a2lem3 10439 . . . . . . . . . . . . . 14 (∅ ∈ ω → (𝐸‘∅) = (2o ·o ∅))
3331, 32ax-mp 5 . . . . . . . . . . . . 13 (𝐸‘∅) = (2o ·o ∅)
34 2on 8518 . . . . . . . . . . . . . 14 2o ∈ On
35 om0 8553 . . . . . . . . . . . . . 14 (2o ∈ On → (2o ·o ∅) = ∅)
3634, 35ax-mp 5 . . . . . . . . . . . . 13 (2o ·o ∅) = ∅
3733, 36eqtri 2762 . . . . . . . . . . . 12 (𝐸‘∅) = ∅
38 f1fun 6806 . . . . . . . . . . . . . 14 (𝐸:ω–1-1→ω → Fun 𝐸)
394, 38ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐸
40 f1dm 6808 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1→ω → dom 𝐸 = ω)
414, 40ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐸 = ω
4231, 41eleqtrri 2837 . . . . . . . . . . . . 13 ∅ ∈ dom 𝐸
43 fvelrn 7095 . . . . . . . . . . . . 13 ((Fun 𝐸 ∧ ∅ ∈ dom 𝐸) → (𝐸‘∅) ∈ ran 𝐸)
4439, 42, 43mp2an 692 . . . . . . . . . . . 12 (𝐸‘∅) ∈ ran 𝐸
4537, 44eqeltrri 2835 . . . . . . . . . . 11 ∅ ∈ ran 𝐸
46 eleq1 2826 . . . . . . . . . . 11 (𝑎 = ∅ → (𝑎 ∈ ran 𝐸 ↔ ∅ ∈ ran 𝐸))
4745, 46mpbiri 258 . . . . . . . . . 10 (𝑎 = ∅ → 𝑎 ∈ ran 𝐸)
4847necon3bi 2964 . . . . . . . . 9 𝑎 ∈ ran 𝐸𝑎 ≠ ∅)
49 nnsuc 7904 . . . . . . . . 9 ((𝑎 ∈ ω ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
5048, 49sylan2 593 . . . . . . . 8 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
51 eleq1 2826 . . . . . . . . . . . . 13 (𝑎 = suc 𝑏 → (𝑎 ∈ ω ↔ suc 𝑏 ∈ ω))
52 eleq1 2826 . . . . . . . . . . . . . 14 (𝑎 = suc 𝑏 → (𝑎 ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸))
5352notbid 318 . . . . . . . . . . . . 13 (𝑎 = suc 𝑏 → (¬ 𝑎 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
5451, 53anbi12d 632 . . . . . . . . . . . 12 (𝑎 = suc 𝑏 → ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ↔ (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸)))
5554anbi1d 631 . . . . . . . . . . 11 (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) ↔ ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω)))
56 simplr 769 . . . . . . . . . . . 12 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → ¬ suc 𝑏 ∈ ran 𝐸)
5721adantl 481 . . . . . . . . . . . 12 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
5856, 57mpbird 257 . . . . . . . . . . 11 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸)
5955, 58biimtrdi 253 . . . . . . . . . 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 454 . . . . . . . 8 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑏 ∈ ran 𝐸)
62 simprr 773 . . . . . . . . 9 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑎 = suc 𝑏)
6362eqcomd 2740 . . . . . . . 8 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → suc 𝑏 = 𝑎)
6450, 61, 63reximssdv 3170 . . . . . . 7 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)
6530, 64impbii 209 . . . . . 6 (∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
6616, 65bitri 275 . . . . 5 (∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
67 f1fn 6805 . . . . . . 7 (𝑆:On–1-1→On → 𝑆 Fn On)
682, 67ax-mp 5 . . . . . 6 𝑆 Fn On
69 fvelimab 6980 . . . . . 6 ((𝑆 Fn On ∧ ran 𝐸 ⊆ On) → (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎))
7068, 9, 69mp2an 692 . . . . 5 (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎)
71 eldif 3972 . . . . 5 (𝑎 ∈ (ω ∖ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
7266, 70, 713bitr4i 303 . . . 4 (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ 𝑎 ∈ (ω ∖ ran 𝐸))
7372eqriv 2731 . . 3 (𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸)
74 f1oeq3 6838 . . 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 230 1 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wrex 3067  cdif 3959  wss 3962  c0 4338  cmpt 5230  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  Oncon0 6385  suc csuc 6387  Fun wfun 6556   Fn wfn 6557  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  ωcom 7886  2oc2o 8498   ·o comu 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509
This theorem is referenced by:  fin1a2lem7  10443
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