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Theorem fin1a2lem6 10349
Description: Lemma for fin1a2 10359. 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 10345 . . 3 𝑆:On–1-1β†’On
3 fin1a2lem.b . . . . 5 𝐸 = (π‘₯ ∈ Ο‰ ↦ (2o Β·o π‘₯))
43fin1a2lem4 10347 . . . 4 𝐸:ω–1-1β†’Ο‰
5 f1f 6742 . . . 4 (𝐸:ω–1-1β†’Ο‰ β†’ 𝐸:Ο‰βŸΆΟ‰)
6 frn 6679 . . . . 5 (𝐸:Ο‰βŸΆΟ‰ β†’ ran 𝐸 βŠ† Ο‰)
7 omsson 7810 . . . . 5 Ο‰ βŠ† On
86, 7sstrdi 3960 . . . 4 (𝐸:Ο‰βŸΆΟ‰ β†’ ran 𝐸 βŠ† On)
94, 5, 8mp2b 10 . . 3 ran 𝐸 βŠ† On
10 f1ores 6802 . . 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 691 . 2 (𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(𝑆 β€œ ran 𝐸)
129sseli 3944 . . . . . . . . 9 (𝑏 ∈ ran 𝐸 β†’ 𝑏 ∈ On)
131fin1a2lem1 10344 . . . . . . . . 9 (𝑏 ∈ On β†’ (π‘†β€˜π‘) = suc 𝑏)
1412, 13syl 17 . . . . . . . 8 (𝑏 ∈ ran 𝐸 β†’ (π‘†β€˜π‘) = suc 𝑏)
1514eqeq1d 2735 . . . . . . 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 3944 . . . . . . . . . . 11 (𝑏 ∈ ran 𝐸 β†’ 𝑏 ∈ Ο‰)
19 peano2 7831 . . . . . . . . . . 11 (𝑏 ∈ Ο‰ β†’ suc 𝑏 ∈ Ο‰)
2018, 19syl 17 . . . . . . . . . 10 (𝑏 ∈ ran 𝐸 β†’ suc 𝑏 ∈ Ο‰)
213fin1a2lem5 10348 . . . . . . . . . . . 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 513 . . . . . . . . 9 (𝑏 ∈ ran 𝐸 β†’ (suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸))
25 eleq1 2822 . . . . . . . . . 10 (suc 𝑏 = π‘Ž β†’ (suc 𝑏 ∈ Ο‰ ↔ π‘Ž ∈ Ο‰))
26 eleq1 2822 . . . . . . . . . . 11 (suc 𝑏 = π‘Ž β†’ (suc 𝑏 ∈ ran 𝐸 ↔ π‘Ž ∈ ran 𝐸))
2726notbid 318 . . . . . . . . . 10 (suc 𝑏 = π‘Ž β†’ (Β¬ suc 𝑏 ∈ ran 𝐸 ↔ Β¬ π‘Ž ∈ ran 𝐸))
2825, 27anbi12d 632 . . . . . . . . 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 3142 . . . . . . 7 (βˆƒπ‘ ∈ ran 𝐸 suc 𝑏 = π‘Ž β†’ (π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸))
31 peano1 7829 . . . . . . . . . . . . . 14 βˆ… ∈ Ο‰
323fin1a2lem3 10346 . . . . . . . . . . . . . 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 8430 . . . . . . . . . . . . . 14 2o ∈ On
35 om0 8467 . . . . . . . . . . . . . 14 (2o ∈ On β†’ (2o Β·o βˆ…) = βˆ…)
3634, 35ax-mp 5 . . . . . . . . . . . . 13 (2o Β·o βˆ…) = βˆ…
3733, 36eqtri 2761 . . . . . . . . . . . 12 (πΈβ€˜βˆ…) = βˆ…
38 f1fun 6744 . . . . . . . . . . . . . 14 (𝐸:ω–1-1β†’Ο‰ β†’ Fun 𝐸)
394, 38ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐸
40 f1dm 6746 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1β†’Ο‰ β†’ dom 𝐸 = Ο‰)
414, 40ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐸 = Ο‰
4231, 41eleqtrri 2833 . . . . . . . . . . . . 13 βˆ… ∈ dom 𝐸
43 fvelrn 7031 . . . . . . . . . . . . 13 ((Fun 𝐸 ∧ βˆ… ∈ dom 𝐸) β†’ (πΈβ€˜βˆ…) ∈ ran 𝐸)
4439, 42, 43mp2an 691 . . . . . . . . . . . 12 (πΈβ€˜βˆ…) ∈ ran 𝐸
4537, 44eqeltrri 2831 . . . . . . . . . . 11 βˆ… ∈ ran 𝐸
46 eleq1 2822 . . . . . . . . . . 11 (π‘Ž = βˆ… β†’ (π‘Ž ∈ ran 𝐸 ↔ βˆ… ∈ ran 𝐸))
4745, 46mpbiri 258 . . . . . . . . . 10 (π‘Ž = βˆ… β†’ π‘Ž ∈ ran 𝐸)
4847necon3bi 2967 . . . . . . . . 9 (Β¬ π‘Ž ∈ ran 𝐸 β†’ π‘Ž β‰  βˆ…)
49 nnsuc 7824 . . . . . . . . 9 ((π‘Ž ∈ Ο‰ ∧ π‘Ž β‰  βˆ…) β†’ βˆƒπ‘ ∈ Ο‰ π‘Ž = suc 𝑏)
5048, 49sylan2 594 . . . . . . . 8 ((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) β†’ βˆƒπ‘ ∈ Ο‰ π‘Ž = suc 𝑏)
51 eleq1 2822 . . . . . . . . . . . . 13 (π‘Ž = suc 𝑏 β†’ (π‘Ž ∈ Ο‰ ↔ suc 𝑏 ∈ Ο‰))
52 eleq1 2822 . . . . . . . . . . . . . 14 (π‘Ž = suc 𝑏 β†’ (π‘Ž ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸))
5352notbid 318 . . . . . . . . . . . . 13 (π‘Ž = suc 𝑏 β†’ (Β¬ π‘Ž ∈ ran 𝐸 ↔ Β¬ suc 𝑏 ∈ ran 𝐸))
5451, 53anbi12d 632 . . . . . . . . . . . 12 (π‘Ž = suc 𝑏 β†’ ((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ↔ (suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸)))
5554anbi1d 631 . . . . . . . . . . 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 768 . . . . . . . . . . . 12 (((suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰) β†’ Β¬ suc 𝑏 ∈ ran 𝐸)
5721adantl 483 . . . . . . . . . . . 12 (((suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰) β†’ (𝑏 ∈ ran 𝐸 ↔ Β¬ suc 𝑏 ∈ ran 𝐸))
5856, 57mpbird 257 . . . . . . . . . . 11 (((suc 𝑏 ∈ Ο‰ ∧ Β¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ Ο‰) β†’ 𝑏 ∈ ran 𝐸)
5955, 58syl6bi 253 . . . . . . . . . 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 456 . . . . . . . 8 (((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ∧ (𝑏 ∈ Ο‰ ∧ π‘Ž = suc 𝑏)) β†’ 𝑏 ∈ ran 𝐸)
62 simprr 772 . . . . . . . . 9 (((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ∧ (𝑏 ∈ Ο‰ ∧ π‘Ž = suc 𝑏)) β†’ π‘Ž = suc 𝑏)
6362eqcomd 2739 . . . . . . . 8 (((π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸) ∧ (𝑏 ∈ Ο‰ ∧ π‘Ž = suc 𝑏)) β†’ suc 𝑏 = π‘Ž)
6450, 61, 63reximssdv 3166 . . . . . . 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 275 . . . . 5 (βˆƒπ‘ ∈ ran 𝐸(π‘†β€˜π‘) = π‘Ž ↔ (π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸))
67 f1fn 6743 . . . . . . 7 (𝑆:On–1-1β†’On β†’ 𝑆 Fn On)
682, 67ax-mp 5 . . . . . 6 𝑆 Fn On
69 fvelimab 6918 . . . . . 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 691 . . . . 5 (π‘Ž ∈ (𝑆 β€œ ran 𝐸) ↔ βˆƒπ‘ ∈ ran 𝐸(π‘†β€˜π‘) = π‘Ž)
71 eldif 3924 . . . . 5 (π‘Ž ∈ (Ο‰ βˆ– ran 𝐸) ↔ (π‘Ž ∈ Ο‰ ∧ Β¬ π‘Ž ∈ ran 𝐸))
7266, 70, 713bitr4i 303 . . . 4 (π‘Ž ∈ (𝑆 β€œ ran 𝐸) ↔ π‘Ž ∈ (Ο‰ βˆ– ran 𝐸))
7372eqriv 2730 . . 3 (𝑆 β€œ ran 𝐸) = (Ο‰ βˆ– ran 𝐸)
74 f1oeq3 6778 . . 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070   βˆ– cdif 3911   βŠ† wss 3914  βˆ…c0 4286   ↦ cmpt 5192  dom cdm 5637  ran crn 5638   β†Ύ cres 5639   β€œ cima 5640  Oncon0 6321  suc csuc 6323  Fun wfun 6494   Fn wfn 6495  βŸΆwf 6496  β€“1-1β†’wf1 6497  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361  Ο‰com 7806  2oc2o 8410   Β·o comu 8414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-oadd 8420  df-omul 8421
This theorem is referenced by:  fin1a2lem7  10350
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