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Theorem alephfp 10044
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 10045 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 10043 . 2 βˆͺ (𝐻 β€œ Ο‰) ∈ ran β„΅
3 isinfcard 10028 . . 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 10026 . . . 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 477 . . 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 10024 . . . . . . . . 9 (𝑧 ∈ On β†’ 𝑧 βŠ† (β„΅β€˜π‘§))
8 alephon 10005 . . . . . . . . . . 11 (β„΅β€˜π‘§) ∈ On
98onirri 6430 . . . . . . . . . 10 Β¬ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§)
10 frfnom 8381 . . . . . . . . . . . . . 14 (rec(β„΅, Ο‰) β†Ύ Ο‰) Fn Ο‰
111fneq1i 6599 . . . . . . . . . . . . . 14 (𝐻 Fn Ο‰ ↔ (rec(β„΅, Ο‰) β†Ύ Ο‰) Fn Ο‰)
1210, 11mpbir 230 . . . . . . . . . . . . 13 𝐻 Fn Ο‰
13 fnfun 6602 . . . . . . . . . . . . 13 (𝐻 Fn Ο‰ β†’ Fun 𝐻)
14 eluniima 7197 . . . . . . . . . . . . 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 10036 . . . . . . . . . . . . . . . 16 ran β„΅ βŠ† On
171alephfplem3 10042 . . . . . . . . . . . . . . . 16 (𝑣 ∈ Ο‰ β†’ (π»β€˜π‘£) ∈ ran β„΅)
1816, 17sselid 3942 . . . . . . . . . . . . . . 15 (𝑣 ∈ Ο‰ β†’ (π»β€˜π‘£) ∈ On)
19 alephord2i 10013 . . . . . . . . . . . . . . 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 10041 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) = (β„΅β€˜(π»β€˜π‘£)))
22 peano2 7827 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ Ο‰ β†’ suc 𝑣 ∈ Ο‰)
23 fnfvelrn 7031 . . . . . . . . . . . . . . . . . . . 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 6631 . . . . . . . . . . . . . . . . . . . 20 (𝐻 Fn Ο‰ β†’ (𝐻 β€œ Ο‰) = ran 𝐻)
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐻 β€œ Ο‰) = ran 𝐻
2724, 26eleqtrrdi 2848 . . . . . . . . . . . . . . . . . 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 2838 . . . . . . . . . . . . . . . 16 (𝑣 ∈ Ο‰ β†’ (β„΅β€˜(π»β€˜π‘£)) ∈ (𝐻 β€œ Ο‰))
30 elssuni 4898 . . . . . . . . . . . . . . . 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 3943 . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . 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 2826 . . . . . . . . . . . 12 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 ∈ (β„΅β€˜π‘§)))
37 eleq2 2826 . . . . . . . . . . . 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 613 . . . . . . . 8 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§)))
42 eloni 6327 . . . . . . . . . 10 (𝑧 ∈ On β†’ Ord 𝑧)
438onordi 6428 . . . . . . . . . 10 Ord (β„΅β€˜π‘§)
44 ordtri4 6354 . . . . . . . . . 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 586 . . . . . . . . 9 (𝑧 ∈ On β†’ (𝑧 = (β„΅β€˜π‘§) ↔ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))))
4645adantr 481 . . . . . . . 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 2748 . . . . . . . 8 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 = βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 = (β„΅β€˜π‘§)))
4948adantl 482 . . . . . . 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 2742 . . . . 5 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ βˆͺ (𝐻 β€œ Ο‰) = 𝑧)
5251fveq2d 6846 . . . 4 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = (β„΅β€˜π‘§))
53 eqeq2 2748 . . . . 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 482 . . . 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 3144 . 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 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3073   βŠ† wss 3910  βˆͺ cuni 4865  ran crn 5634   β†Ύ cres 5635   β€œ cima 5636  Ord word 6316  Oncon0 6317  suc csuc 6319  Fun wfun 6490   Fn wfn 6491  β€˜cfv 6496  Ο‰com 7802  reccrdg 8355  cardccrd 9871  β„΅cale 9872
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-har 9493  df-card 9875  df-aleph 9876
This theorem is referenced by:  alephfp2  10045
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