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Theorem fin1a2lem6 9224
Description: Lemma for fin1a2 9234. 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 9220 . . 3 𝑆:On–1-1→On
3 fin1a2lem.b . . . . 5 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
43fin1a2lem4 9222 . . . 4 𝐸:ω–1-1→ω
5 f1f 6099 . . . 4 (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
6 frn 6051 . . . . 5 (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
7 omsson 7066 . . . . 5 ω ⊆ On
86, 7syl6ss 3613 . . . 4 (𝐸:ω⟶ω → ran 𝐸 ⊆ On)
94, 5, 8mp2b 10 . . 3 ran 𝐸 ⊆ On
10 f1ores 6149 . . 3 ((𝑆:On–1-1→On ∧ ran 𝐸 ⊆ On) → (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸))
112, 9, 10mp2an 708 . 2 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(𝑆 “ ran 𝐸)
129sseli 3597 . . . . . . . . 9 (𝑏 ∈ ran 𝐸𝑏 ∈ On)
131fin1a2lem1 9219 . . . . . . . . 9 (𝑏 ∈ On → (𝑆𝑏) = suc 𝑏)
1412, 13syl 17 . . . . . . . 8 (𝑏 ∈ ran 𝐸 → (𝑆𝑏) = suc 𝑏)
1514eqeq1d 2623 . . . . . . 7 (𝑏 ∈ ran 𝐸 → ((𝑆𝑏) = 𝑎 ↔ suc 𝑏 = 𝑎))
1615rexbiia 3038 . . . . . 6 (∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎 ↔ ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)
174, 5, 6mp2b 10 . . . . . . . . . . . 12 ran 𝐸 ⊆ ω
1817sseli 3597 . . . . . . . . . . 11 (𝑏 ∈ ran 𝐸𝑏 ∈ ω)
19 peano2 7083 . . . . . . . . . . 11 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
2018, 19syl 17 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 → suc 𝑏 ∈ ω)
213fin1a2lem5 9223 . . . . . . . . . . . 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 554 . . . . . . . . 9 (𝑏 ∈ ran 𝐸 → (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸))
25 eleq1 2688 . . . . . . . . . 10 (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ω ↔ 𝑎 ∈ ω))
26 eleq1 2688 . . . . . . . . . . 11 (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ran 𝐸𝑎 ∈ ran 𝐸))
2726notbid 308 . . . . . . . . . 10 (suc 𝑏 = 𝑎 → (¬ suc 𝑏 ∈ ran 𝐸 ↔ ¬ 𝑎 ∈ ran 𝐸))
2825, 27anbi12d 747 . . . . . . . . 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 3025 . . . . . . 7 (∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
31 peano1 7082 . . . . . . . . . . . . . 14 ∅ ∈ ω
323fin1a2lem3 9221 . . . . . . . . . . . . . 14 (∅ ∈ ω → (𝐸‘∅) = (2𝑜 ·𝑜 ∅))
3331, 32ax-mp 5 . . . . . . . . . . . . 13 (𝐸‘∅) = (2𝑜 ·𝑜 ∅)
34 om0x 7596 . . . . . . . . . . . . 13 (2𝑜 ·𝑜 ∅) = ∅
3533, 34eqtri 2643 . . . . . . . . . . . 12 (𝐸‘∅) = ∅
36 f1fun 6101 . . . . . . . . . . . . . 14 (𝐸:ω–1-1→ω → Fun 𝐸)
374, 36ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐸
38 f1dm 6103 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1→ω → dom 𝐸 = ω)
394, 38ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐸 = ω
4031, 39eleqtrri 2699 . . . . . . . . . . . . 13 ∅ ∈ dom 𝐸
41 fvelrn 6350 . . . . . . . . . . . . 13 ((Fun 𝐸 ∧ ∅ ∈ dom 𝐸) → (𝐸‘∅) ∈ ran 𝐸)
4237, 40, 41mp2an 708 . . . . . . . . . . . 12 (𝐸‘∅) ∈ ran 𝐸
4335, 42eqeltrri 2697 . . . . . . . . . . 11 ∅ ∈ ran 𝐸
44 eleq1 2688 . . . . . . . . . . 11 (𝑎 = ∅ → (𝑎 ∈ ran 𝐸 ↔ ∅ ∈ ran 𝐸))
4543, 44mpbiri 248 . . . . . . . . . 10 (𝑎 = ∅ → 𝑎 ∈ ran 𝐸)
4645necon3bi 2819 . . . . . . . . 9 𝑎 ∈ ran 𝐸𝑎 ≠ ∅)
47 nnsuc 7079 . . . . . . . . 9 ((𝑎 ∈ ω ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
4846, 47sylan2 491 . . . . . . . 8 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
49 eleq1 2688 . . . . . . . . . . . . . . . 16 (𝑎 = suc 𝑏 → (𝑎 ∈ ω ↔ suc 𝑏 ∈ ω))
50 eleq1 2688 . . . . . . . . . . . . . . . . 17 (𝑎 = suc 𝑏 → (𝑎 ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸))
5150notbid 308 . . . . . . . . . . . . . . . 16 (𝑎 = suc 𝑏 → (¬ 𝑎 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
5249, 51anbi12d 747 . . . . . . . . . . . . . . 15 (𝑎 = suc 𝑏 → ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ↔ (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸)))
5352anbi1d 741 . . . . . . . . . . . . . 14 (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) ↔ ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω)))
54 simplr 792 . . . . . . . . . . . . . . 15 (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → ¬ suc 𝑏 ∈ ran 𝐸)
5521adantl 482 . . . . . . . . . . . . . . 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 649 . . . . . . . . . . 11 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑏 ∈ ran 𝐸)
60 simprr 796 . . . . . . . . . . . 12 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑎 = suc 𝑏)
6160eqcomd 2627 . . . . . . . . . . 11 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → suc 𝑏 = 𝑎)
6259, 61jca 554 . . . . . . . . . 10 (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → (𝑏 ∈ ran 𝐸 ∧ suc 𝑏 = 𝑎))
6362ex 450 . . . . . . . . 9 ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ((𝑏 ∈ ω ∧ 𝑎 = suc 𝑏) → (𝑏 ∈ ran 𝐸 ∧ suc 𝑏 = 𝑎)))
6463reximdv2 3013 . . . . . . . 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 6100 . . . . . . 7 (𝑆:On–1-1→On → 𝑆 Fn On)
692, 68ax-mp 5 . . . . . 6 𝑆 Fn On
70 fvelimab 6251 . . . . . 6 ((𝑆 Fn On ∧ ran 𝐸 ⊆ On) → (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎))
7169, 9, 70mp2an 708 . . . . 5 (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆𝑏) = 𝑎)
72 eldif 3582 . . . . 5 (𝑎 ∈ (ω ∖ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
7367, 71, 723bitr4i 292 . . . 4 (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ 𝑎 ∈ (ω ∖ ran 𝐸))
7473eqriv 2618 . . 3 (𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸)
75 f1oeq3 6127 . . 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 384   = wceq 1482  wcel 1989  wne 2793  wrex 2912  cdif 3569  wss 3572  c0 3913  cmpt 4727  dom cdm 5112  ran crn 5113  cres 5114  cima 5115  Oncon0 5721  suc csuc 5723  Fun wfun 5880   Fn wfn 5881  wf 5882  1-1wf1 5883  1-1-ontowf1o 5885  cfv 5886  (class class class)co 6647  ωcom 7062  2𝑜c2o 7551   ·𝑜 comu 7555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-omul 7562
This theorem is referenced by:  fin1a2lem7  9225
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