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Theorem fin1a2lem6 10399
Description: Lemma for fin1a2 10409. Establish that Ο‰ can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypotheses
Ref Expression
fin1a2lem.b 𝐸 = (π‘₯ ∈ Ο‰ ↦ (2o Β·o π‘₯))
fin1a2lem.aa 𝑆 = (π‘₯ ∈ On ↦ suc π‘₯)
Assertion
Ref Expression
fin1a2lem6 (𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(Ο‰ βˆ– ran 𝐸)

Proof of Theorem fin1a2lem6
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.aa . . . 4 𝑆 = (π‘₯ ∈ On ↦ suc π‘₯)
21fin1a2lem2 10395 . . 3 𝑆:On–1-1β†’On
3 fin1a2lem.b . . . . 5 𝐸 = (π‘₯ ∈ Ο‰ ↦ (2o Β·o π‘₯))
43fin1a2lem4 10397 . . . 4 𝐸:ω–1-1β†’Ο‰
5 f1f 6787 . . . 4 (𝐸:ω–1-1β†’Ο‰ β†’ 𝐸:Ο‰βŸΆΟ‰)
6 frn 6724 . . . . 5 (𝐸:Ο‰βŸΆΟ‰ β†’ ran 𝐸 βŠ† Ο‰)
7 omsson 7858 . . . . 5 Ο‰ βŠ† On
86, 7sstrdi 3994 . . . 4 (𝐸:Ο‰βŸΆΟ‰ β†’ ran 𝐸 βŠ† On)
94, 5, 8mp2b 10 . . 3 ran 𝐸 βŠ† On
10 f1ores 6847 . . 3 ((𝑆:On–1-1β†’On ∧ ran 𝐸 βŠ† On) β†’ (𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(𝑆 β€œ ran 𝐸))
112, 9, 10mp2an 690 . 2 (𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(𝑆 β€œ ran 𝐸)
129sseli 3978 . . . . . . . . 9 (𝑏 ∈ ran 𝐸 β†’ 𝑏 ∈ On)
131fin1a2lem1 10394 . . . . . . . . 9 (𝑏 ∈ On β†’ (π‘†β€˜π‘) = suc 𝑏)
1412, 13syl 17 . . . . . . . 8 (𝑏 ∈ ran 𝐸 β†’ (π‘†β€˜π‘) = suc 𝑏)
1514eqeq1d 2734 . . . . . . 7 (𝑏 ∈ ran 𝐸 β†’ ((π‘†β€˜π‘) = π‘Ž ↔ suc 𝑏 = π‘Ž))
1615rexbiia 3092 . . . . . 6 (βˆƒπ‘ ∈ ran 𝐸(π‘†β€˜π‘) = π‘Ž ↔ βˆƒπ‘ ∈ ran 𝐸 suc 𝑏 = π‘Ž)
174, 5, 6mp2b 10 . . . . . . . . . . . 12 ran 𝐸 βŠ† Ο‰
1817sseli 3978 . . . . . . . . . . 11 (𝑏 ∈ ran 𝐸 β†’ 𝑏 ∈ Ο‰)
19 peano2 7880 . . . . . . . . . . 11 (𝑏 ∈ Ο‰ β†’ suc 𝑏 ∈ Ο‰)
2018, 19syl 17 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 β†’ suc 𝑏 ∈ Ο‰)
213fin1a2lem5 10398 . . . . . . . . . . . 12 (𝑏 ∈ Ο‰ β†’ (𝑏 ∈ ran 𝐸 ↔ Β¬ suc 𝑏 ∈ ran 𝐸))
2221biimpd 228 . . . . . . . . . . 11 (𝑏 ∈ Ο‰ β†’ (𝑏 ∈ ran 𝐸 β†’ Β¬ suc 𝑏 ∈ ran 𝐸))
2318, 22mpcom 38 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 β†’ Β¬ suc 𝑏 ∈ ran 𝐸)
2420, 23jca 512 . . . . . . . . 9 (𝑏 ∈ ran 𝐸 β†’ (suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸))
25 eleq1 2821 . . . . . . . . . 10 (suc 𝑏 = π‘Ž β†’ (suc 𝑏 ∈ Ο‰ ↔ π‘Ž ∈ Ο‰))
26 eleq1 2821 . . . . . . . . . . 11 (suc 𝑏 = π‘Ž β†’ (suc 𝑏 ∈ ran 𝐸 ↔ π‘Ž ∈ ran 𝐸))
2726notbid 317 . . . . . . . . . 10 (suc 𝑏 = π‘Ž β†’ (Β¬ suc 𝑏 ∈ ran 𝐸 ↔ Β¬ π‘Ž ∈ ran 𝐸))
2825, 27anbi12d 631 . . . . . . . . 9 (suc 𝑏 = π‘Ž β†’ ((suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸) ↔ (π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸)))
2924, 28syl5ibcom 244 . . . . . . . 8 (𝑏 ∈ ran 𝐸 β†’ (suc 𝑏 = π‘Ž β†’ (π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸)))
3029rexlimiv 3148 . . . . . . 7 (βˆƒπ‘ ∈ ran 𝐸 suc 𝑏 = π‘Ž β†’ (π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸))
31 peano1 7878 . . . . . . . . . . . . . 14 βˆ… ∈ Ο‰
323fin1a2lem3 10396 . . . . . . . . . . . . . 14 (βˆ… ∈ Ο‰ β†’ (πΈβ€˜βˆ…) = (2o Β·o βˆ…))
3331, 32ax-mp 5 . . . . . . . . . . . . 13 (πΈβ€˜βˆ…) = (2o Β·o βˆ…)
34 2on 8479 . . . . . . . . . . . . . 14 2o ∈ On
35 om0 8516 . . . . . . . . . . . . . 14 (2o ∈ On β†’ (2o Β·o βˆ…) = βˆ…)
3634, 35ax-mp 5 . . . . . . . . . . . . 13 (2o Β·o βˆ…) = βˆ…
3733, 36eqtri 2760 . . . . . . . . . . . 12 (πΈβ€˜βˆ…) = βˆ…
38 f1fun 6789 . . . . . . . . . . . . . 14 (𝐸:ω–1-1β†’Ο‰ β†’ Fun 𝐸)
394, 38ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐸
40 f1dm 6791 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1β†’Ο‰ β†’ dom 𝐸 = Ο‰)
414, 40ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐸 = Ο‰
4231, 41eleqtrri 2832 . . . . . . . . . . . . 13 βˆ… ∈ dom 𝐸
43 fvelrn 7078 . . . . . . . . . . . . 13 ((Fun 𝐸 ∧ βˆ… ∈ dom 𝐸) β†’ (πΈβ€˜βˆ…) ∈ ran 𝐸)
4439, 42, 43mp2an 690 . . . . . . . . . . . 12 (πΈβ€˜βˆ…) ∈ ran 𝐸
4537, 44eqeltrri 2830 . . . . . . . . . . 11 βˆ… ∈ ran 𝐸
46 eleq1 2821 . . . . . . . . . . 11 (π‘Ž = βˆ… β†’ (π‘Ž ∈ ran 𝐸 ↔ βˆ… ∈ ran 𝐸))
4745, 46mpbiri 257 . . . . . . . . . 10 (π‘Ž = βˆ… β†’ π‘Ž ∈ ran 𝐸)
4847necon3bi 2967 . . . . . . . . 9 (Β¬ π‘Ž ∈ ran 𝐸 β†’ π‘Ž β‰  βˆ…)
49 nnsuc 7872 . . . . . . . . 9 ((π‘Ž ∈ Ο‰ ∧ π‘Ž β‰  βˆ…) β†’ βˆƒπ‘ ∈ Ο‰ π‘Ž = suc 𝑏)
5048, 49sylan2 593 . . . . . . . 8 ((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) β†’ βˆƒπ‘ ∈ Ο‰ π‘Ž = suc 𝑏)
51 eleq1 2821 . . . . . . . . . . . . 13 (π‘Ž = suc 𝑏 β†’ (π‘Ž ∈ Ο‰ ↔ suc 𝑏 ∈ Ο‰))
52 eleq1 2821 . . . . . . . . . . . . . 14 (π‘Ž = suc 𝑏 β†’ (π‘Ž ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸))
5352notbid 317 . . . . . . . . . . . . 13 (π‘Ž = suc 𝑏 β†’ (Β¬ π‘Ž ∈ ran 𝐸 ↔ Β¬ suc 𝑏 ∈ ran 𝐸))
5451, 53anbi12d 631 . . . . . . . . . . . 12 (π‘Ž = suc 𝑏 β†’ ((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ↔ (suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸)))
5554anbi1d 630 . . . . . . . . . . 11 (π‘Ž = suc 𝑏 β†’ (((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰) ↔ ((suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰)))
56 simplr 767 . . . . . . . . . . . 12 (((suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰) β†’ Β¬ suc 𝑏 ∈ ran 𝐸)
5721adantl 482 . . . . . . . . . . . 12 (((suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰) β†’ (𝑏 ∈ ran 𝐸 ↔ Β¬ suc 𝑏 ∈ ran 𝐸))
5856, 57mpbird 256 . . . . . . . . . . 11 (((suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰) β†’ 𝑏 ∈ ran 𝐸)
5955, 58syl6bi 252 . . . . . . . . . 10 (π‘Ž = suc 𝑏 β†’ (((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰) β†’ 𝑏 ∈ ran 𝐸))
6059com12 32 . . . . . . . . 9 (((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰) β†’ (π‘Ž = suc 𝑏 β†’ 𝑏 ∈ ran 𝐸))
6160impr 455 . . . . . . . 8 (((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ∧ (𝑏 ∈ Ο‰ ∧ π‘Ž = suc 𝑏)) β†’ 𝑏 ∈ ran 𝐸)
62 simprr 771 . . . . . . . . 9 (((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ∧ (𝑏 ∈ Ο‰ ∧ π‘Ž = suc 𝑏)) β†’ π‘Ž = suc 𝑏)
6362eqcomd 2738 . . . . . . . 8 (((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ∧ (𝑏 ∈ Ο‰ ∧ π‘Ž = suc 𝑏)) β†’ suc 𝑏 = π‘Ž)
6450, 61, 63reximssdv 3172 . . . . . . 7 ((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) β†’ βˆƒπ‘ ∈ ran 𝐸 suc 𝑏 = π‘Ž)
6530, 64impbii 208 . . . . . 6 (βˆƒπ‘ ∈ ran 𝐸 suc 𝑏 = π‘Ž ↔ (π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸))
6616, 65bitri 274 . . . . 5 (βˆƒπ‘ ∈ ran 𝐸(π‘†β€˜π‘) = π‘Ž ↔ (π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸))
67 f1fn 6788 . . . . . . 7 (𝑆:On–1-1β†’On β†’ 𝑆 Fn On)
682, 67ax-mp 5 . . . . . 6 𝑆 Fn On
69 fvelimab 6964 . . . . . 6 ((𝑆 Fn On ∧ ran 𝐸 βŠ† On) β†’ (π‘Ž ∈ (𝑆 β€œ ran 𝐸) ↔ βˆƒπ‘ ∈ ran 𝐸(π‘†β€˜π‘) = π‘Ž))
7068, 9, 69mp2an 690 . . . . 5 (π‘Ž ∈ (𝑆 β€œ ran 𝐸) ↔ βˆƒπ‘ ∈ ran 𝐸(π‘†β€˜π‘) = π‘Ž)
71 eldif 3958 . . . . 5 (π‘Ž ∈ (Ο‰ βˆ– ran 𝐸) ↔ (π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸))
7266, 70, 713bitr4i 302 . . . 4 (π‘Ž ∈ (𝑆 β€œ ran 𝐸) ↔ π‘Ž ∈ (Ο‰ βˆ– ran 𝐸))
7372eqriv 2729 . . 3 (𝑆 β€œ ran 𝐸) = (Ο‰ βˆ– ran 𝐸)
74 f1oeq3 6823 . . 3 ((𝑆 β€œ ran 𝐸) = (Ο‰ βˆ– ran 𝐸) β†’ ((𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(𝑆 β€œ ran 𝐸) ↔ (𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(Ο‰ βˆ– ran 𝐸)))
7573, 74ax-mp 5 . 2 ((𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(𝑆 β€œ ran 𝐸) ↔ (𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(Ο‰ βˆ– ran 𝐸))
7611, 75mpbi 229 1 (𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(Ο‰ βˆ– ran 𝐸)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   βˆ– cdif 3945   βŠ† wss 3948  βˆ…c0 4322   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679  Oncon0 6364  suc csuc 6366  Fun wfun 6537   Fn wfn 6538  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7408  Ο‰com 7854  2oc2o 8459   Β·o comu 8463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-2o 8466  df-oadd 8469  df-omul 8470
This theorem is referenced by:  fin1a2lem7  10400
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