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Theorem alephfp 10123
Description: The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 10124 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
alephfplem.1 𝐻 = (rec(β„΅, Ο‰) β†Ύ Ο‰)
Assertion
Ref Expression
alephfp (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰)
Proof of Theorem alephfp
Dummy variables 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephfplem.1 . . 3 𝐻 = (rec(β„΅, Ο‰) β†Ύ Ο‰)
21alephfplem4 10122 . 2 βˆͺ (𝐻 β€œ Ο‰) ∈ ran β„΅
3 isinfcard 10107 . . 3 ((Ο‰ βŠ† βˆͺ (𝐻 β€œ Ο‰) ∧ (cardβ€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰)) ↔ βˆͺ (𝐻 β€œ Ο‰) ∈ ran β„΅)
4 cardalephex 10105 . . . 4 (Ο‰ βŠ† βˆͺ (𝐻 β€œ Ο‰) β†’ ((cardβ€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰) ↔ βˆƒπ‘§ ∈ On βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)))
54biimpa 476 . . 3 ((Ο‰ βŠ† βˆͺ (𝐻 β€œ Ο‰) ∧ (cardβ€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰)) β†’ βˆƒπ‘§ ∈ On βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§))
63, 5sylbir 234 . 2 (βˆͺ (𝐻 β€œ Ο‰) ∈ ran β„΅ β†’ βˆƒπ‘§ ∈ On βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§))
7 alephle 10103 . . . . . . . . 9 (𝑧 ∈ On β†’ 𝑧 βŠ† (β„΅β€˜π‘§))
8 alephon 10084 . . . . . . . . . . 11 (β„΅β€˜π‘§) ∈ On
98onirri 6476 . . . . . . . . . 10 Β¬ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§)
10 frfnom 8449 . . . . . . . . . . . . . 14 (rec(β„΅, Ο‰) β†Ύ Ο‰) Fn Ο‰
111fneq1i 6645 . . . . . . . . . . . . . 14 (𝐻 Fn Ο‰ ↔ (rec(β„΅, Ο‰) β†Ύ Ο‰) Fn Ο‰)
1210, 11mpbir 230 . . . . . . . . . . . . 13 𝐻 Fn Ο‰
13 fnfun 6648 . . . . . . . . . . . . 13 (𝐻 Fn Ο‰ β†’ Fun 𝐻)
14 eluniima 7254 . . . . . . . . . . . . 13 (Fun 𝐻 β†’ (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ βˆƒπ‘£ ∈ Ο‰ 𝑧 ∈ (π»β€˜π‘£)))
1512, 13, 14mp2b 10 . . . . . . . . . . . 12 (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ βˆƒπ‘£ ∈ Ο‰ 𝑧 ∈ (π»β€˜π‘£))
16 alephsson 10115 . . . . . . . . . . . . . . . 16 ran β„΅ βŠ† On
171alephfplem3 10121 . . . . . . . . . . . . . . . 16 (𝑣 ∈ Ο‰ β†’ (π»β€˜π‘£) ∈ ran β„΅)
1816, 17sselid 3976 . . . . . . . . . . . . . . 15 (𝑣 ∈ Ο‰ β†’ (π»β€˜π‘£) ∈ On)
19 alephord2i 10092 . . . . . . . . . . . . . . 15 ((π»β€˜π‘£) ∈ On β†’ (𝑧 ∈ (π»β€˜π‘£) β†’ (β„΅β€˜π‘§) ∈ (β„΅β€˜(π»β€˜π‘£))))
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝑣 ∈ Ο‰ β†’ (𝑧 ∈ (π»β€˜π‘£) β†’ (β„΅β€˜π‘§) ∈ (β„΅β€˜(π»β€˜π‘£))))
211alephfplem2 10120 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) = (β„΅β€˜(π»β€˜π‘£)))
22 peano2 7890 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ Ο‰ β†’ suc 𝑣 ∈ Ο‰)
23 fnfvelrn 7084 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 Fn Ο‰ ∧ suc 𝑣 ∈ Ο‰) β†’ (π»β€˜suc 𝑣) ∈ ran 𝐻)
2412, 23mpan 689 . . . . . . . . . . . . . . . . . . 19 (suc 𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) ∈ ran 𝐻)
25 fnima 6679 . . . . . . . . . . . . . . . . . . . 20 (𝐻 Fn Ο‰ β†’ (𝐻 β€œ Ο‰) = ran 𝐻)
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐻 β€œ Ο‰) = ran 𝐻
2724, 26eleqtrrdi 2839 . . . . . . . . . . . . . . . . . 18 (suc 𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) ∈ (𝐻 β€œ Ο‰))
2822, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) ∈ (𝐻 β€œ Ο‰))
2921, 28eqeltrrd 2829 . . . . . . . . . . . . . . . 16 (𝑣 ∈ Ο‰ β†’ (β„΅β€˜(π»β€˜π‘£)) ∈ (𝐻 β€œ Ο‰))
30 elssuni 4935 . . . . . . . . . . . . . . . 16 ((β„΅β€˜(π»β€˜π‘£)) ∈ (𝐻 β€œ Ο‰) β†’ (β„΅β€˜(π»β€˜π‘£)) βŠ† βˆͺ (𝐻 β€œ Ο‰))
3129, 30syl 17 . . . . . . . . . . . . . . 15 (𝑣 ∈ Ο‰ β†’ (β„΅β€˜(π»β€˜π‘£)) βŠ† βˆͺ (𝐻 β€œ Ο‰))
3231sseld 3977 . . . . . . . . . . . . . 14 (𝑣 ∈ Ο‰ β†’ ((β„΅β€˜π‘§) ∈ (β„΅β€˜(π»β€˜π‘£)) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰)))
3320, 32syld 47 . . . . . . . . . . . . 13 (𝑣 ∈ Ο‰ β†’ (𝑧 ∈ (π»β€˜π‘£) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰)))
3433rexlimiv 3143 . . . . . . . . . . . 12 (βˆƒπ‘£ ∈ Ο‰ 𝑧 ∈ (π»β€˜π‘£) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰))
3515, 34sylbi 216 . . . . . . . . . . 11 (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰))
36 eleq2 2817 . . . . . . . . . . . 12 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 ∈ (β„΅β€˜π‘§)))
37 eleq2 2817 . . . . . . . . . . . 12 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ ((β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§)))
3836, 37imbi12d 344 . . . . . . . . . . 11 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ ((𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰)) ↔ (𝑧 ∈ (β„΅β€˜π‘§) β†’ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§))))
3935, 38mpbii 232 . . . . . . . . . 10 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 ∈ (β„΅β€˜π‘§) β†’ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§)))
409, 39mtoi 198 . . . . . . . . 9 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))
417, 40anim12i 612 . . . . . . . 8 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§)))
42 eloni 6373 . . . . . . . . . 10 (𝑧 ∈ On β†’ Ord 𝑧)
438onordi 6474 . . . . . . . . . 10 Ord (β„΅β€˜π‘§)
44 ordtri4 6400 . . . . . . . . . 10 ((Ord 𝑧 ∧ Ord (β„΅β€˜π‘§)) β†’ (𝑧 = (β„΅β€˜π‘§) ↔ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))))
4542, 43, 44sylancl 585 . . . . . . . . 9 (𝑧 ∈ On β†’ (𝑧 = (β„΅β€˜π‘§) ↔ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))))
4645adantr 480 . . . . . . . 8 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (𝑧 = (β„΅β€˜π‘§) ↔ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))))
4741, 46mpbird 257 . . . . . . 7 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ 𝑧 = (β„΅β€˜π‘§))
48 eqeq2 2739 . . . . . . . 8 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 = βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 = (β„΅β€˜π‘§)))
4948adantl 481 . . . . . . 7 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (𝑧 = βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 = (β„΅β€˜π‘§)))
5047, 49mpbird 257 . . . . . 6 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ 𝑧 = βˆͺ (𝐻 β€œ Ο‰))
5150eqcomd 2733 . . . . 5 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ βˆͺ (𝐻 β€œ Ο‰) = 𝑧)
5251fveq2d 6895 . . . 4 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = (β„΅β€˜π‘§))
53 eqeq2 2739 . . . . 5 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ ((β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰) ↔ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = (β„΅β€˜π‘§)))
5453adantl 481 . . . 4 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ ((β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰) ↔ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = (β„΅β€˜π‘§)))
5552, 54mpbird 257 . . 3 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰))
5655rexlimiva 3142 . 2 (βˆƒπ‘§ ∈ On βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰))
572, 6, 56mp2b 10 1 (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3065   βŠ† wss 3944  βˆͺ cuni 4903  ran crn 5673   β†Ύ cres 5674   β€œ cima 5675  Ord word 6362  Oncon0 6363  suc csuc 6365  Fun wfun 6536   Fn wfn 6537  β€˜cfv 6542  Ο‰com 7864  reccrdg 8423  cardccrd 9950  β„΅cale 9951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9525  df-har 9572  df-card 9954  df-aleph 9955
This theorem is referenced by:  alephfp2  10124
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