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Theorem alephfp 10131
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 10132 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 10130 . 2 βˆͺ (𝐻 β€œ Ο‰) ∈ ran β„΅
3 isinfcard 10115 . . 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 10113 . . . 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 475 . . 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 10111 . . . . . . . . 9 (𝑧 ∈ On β†’ 𝑧 βŠ† (β„΅β€˜π‘§))
8 alephon 10092 . . . . . . . . . . 11 (β„΅β€˜π‘§) ∈ On
98onirri 6482 . . . . . . . . . 10 Β¬ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§)
10 frfnom 8454 . . . . . . . . . . . . . 14 (rec(β„΅, Ο‰) β†Ύ Ο‰) Fn Ο‰
111fneq1i 6650 . . . . . . . . . . . . . 14 (𝐻 Fn Ο‰ ↔ (rec(β„΅, Ο‰) β†Ύ Ο‰) Fn Ο‰)
1210, 11mpbir 230 . . . . . . . . . . . . 13 𝐻 Fn Ο‰
13 fnfun 6653 . . . . . . . . . . . . 13 (𝐻 Fn Ο‰ β†’ Fun 𝐻)
14 eluniima 7258 . . . . . . . . . . . . 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 10123 . . . . . . . . . . . . . . . 16 ran β„΅ βŠ† On
171alephfplem3 10129 . . . . . . . . . . . . . . . 16 (𝑣 ∈ Ο‰ β†’ (π»β€˜π‘£) ∈ ran β„΅)
1816, 17sselid 3975 . . . . . . . . . . . . . . 15 (𝑣 ∈ Ο‰ β†’ (π»β€˜π‘£) ∈ On)
19 alephord2i 10100 . . . . . . . . . . . . . . 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 10128 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) = (β„΅β€˜(π»β€˜π‘£)))
22 peano2 7895 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ Ο‰ β†’ suc 𝑣 ∈ Ο‰)
23 fnfvelrn 7087 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 Fn Ο‰ ∧ suc 𝑣 ∈ Ο‰) β†’ (π»β€˜suc 𝑣) ∈ ran 𝐻)
2412, 23mpan 688 . . . . . . . . . . . . . . . . . . 19 (suc 𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) ∈ ran 𝐻)
25 fnima 6684 . . . . . . . . . . . . . . . . . . . 20 (𝐻 Fn Ο‰ β†’ (𝐻 β€œ Ο‰) = ran 𝐻)
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐻 β€œ Ο‰) = ran 𝐻
2724, 26eleqtrrdi 2836 . . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . 16 (𝑣 ∈ Ο‰ β†’ (β„΅β€˜(π»β€˜π‘£)) ∈ (𝐻 β€œ Ο‰))
30 elssuni 4940 . . . . . . . . . . . . . . . 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 3976 . . . . . . . . . . . . . 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 3138 . . . . . . . . . . . 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 2814 . . . . . . . . . . . 12 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 ∈ (β„΅β€˜π‘§)))
37 eleq2 2814 . . . . . . . . . . . 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 343 . . . . . . . . . . 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 611 . . . . . . . 8 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§)))
42 eloni 6379 . . . . . . . . . 10 (𝑧 ∈ On β†’ Ord 𝑧)
438onordi 6480 . . . . . . . . . 10 Ord (β„΅β€˜π‘§)
44 ordtri4 6406 . . . . . . . . . 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 584 . . . . . . . . 9 (𝑧 ∈ On β†’ (𝑧 = (β„΅β€˜π‘§) ↔ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))))
4645adantr 479 . . . . . . . 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 256 . . . . . . 7 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ 𝑧 = (β„΅β€˜π‘§))
48 eqeq2 2737 . . . . . . . 8 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 = βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 = (β„΅β€˜π‘§)))
4948adantl 480 . . . . . . 7 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (𝑧 = βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 = (β„΅β€˜π‘§)))
5047, 49mpbird 256 . . . . . 6 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ 𝑧 = βˆͺ (𝐻 β€œ Ο‰))
5150eqcomd 2731 . . . . 5 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ βˆͺ (𝐻 β€œ Ο‰) = 𝑧)
5251fveq2d 6898 . . . 4 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = (β„΅β€˜π‘§))
53 eqeq2 2737 . . . . 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 480 . . . 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 256 . . 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 3137 . 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 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060   βŠ† wss 3945  βˆͺ cuni 4908  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680  Ord word 6368  Oncon0 6369  suc csuc 6371  Fun wfun 6541   Fn wfn 6542  β€˜cfv 6547  Ο‰com 7869  reccrdg 8428  cardccrd 9958  β„΅cale 9959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7373  df-ov 7420  df-om 7870  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-oi 9533  df-har 9580  df-card 9962  df-aleph 9963
This theorem is referenced by:  alephfp2  10132
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