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Theorem abianfp 6487
 Description: "A most fundamental fixed point theorem" of Alexander Abian (1923-1999), apparently proved in 1998. Let , , ,... be the iterates of . The theorem reads (using our variable names): "Let be a mapping from a set into itself. Then has a fixed point if and only if: There exists an element of such that for every ordinal , is an element of , and if is not a fixed point of then the 's are all distinct for every ordinal ." See df-rdg 6439 for the operation. The proof's key idea is to assume that does not have a fixed point, then use the Axiom of Replacement in the form of f1dmex 5767 to derive that the class of all ordinal numbers exists, contradicting onprc 4592. Our version of this theorem does not require the hypothesis that be a mapping. Reference: http://us2.metamath.org:88/abian-themostfixed.html. For an application of this theorem, see http://groups.google.com/group/sci.stat.math/msg/1737ee1133c24aeb for its use in a proof of Tarski's fixed point theorem. (Contributed by NM, 5-Sep-2004.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
abianfp.1
abianfp.2
Assertion
Ref Expression
abianfp
Distinct variable groups:   ,,   ,,,   ,,
Allowed substitution hints:   (,)   ()   (,)

Proof of Theorem abianfp
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 abianfp.1 . . . . . . . . . . 11
2 abianfp.2 . . . . . . . . . . 11
31, 2abianfplem 6486 . . . . . . . . . 10
43imp 418 . . . . . . . . 9
54eleq1d 2362 . . . . . . . 8
65biimprd 214 . . . . . . 7
7 fveq2 5541 . . . . . . . . . . . 12
8 id 19 . . . . . . . . . . . 12
97, 8eqeq12d 2310 . . . . . . . . . . 11
109biimprcd 216 . . . . . . . . . 10
113, 10sylcom 25 . . . . . . . . 9
1211imp 418 . . . . . . . 8
1312pm2.24d 135 . . . . . . 7
146, 13jctird 528 . . . . . 6
1514ex 423 . . . . 5
1615com13 74 . . . 4
1716ralrimdv 2645 . . 3
1817reximia 2661 . 2
19 onprc 4592 . . . . 5
20 r19.26 2688 . . . . . 6
21 fveq2 5541 . . . . . . . . . . . . . . . . 17
22 id 19 . . . . . . . . . . . . . . . . 17
2321, 22eqeq12d 2310 . . . . . . . . . . . . . . . 16
2423notbid 285 . . . . . . . . . . . . . . 15
2524rspccv 2894 . . . . . . . . . . . . . 14
2625imim1d 69 . . . . . . . . . . . . 13
2726ralimdv 2635 . . . . . . . . . . . 12
28 ralim 2627 . . . . . . . . . . . 12
2927, 28syl6 29 . . . . . . . . . . 11
3029imp 418 . . . . . . . . . 10
3130com12 27 . . . . . . . . 9
32 rdgfnon 6447 . . . . . . . . . . . . . . 15
332fneq1i 5354 . . . . . . . . . . . . . . 15
3432, 33mpbir 200 . . . . . . . . . . . . . 14
35 ffnfv 5701 . . . . . . . . . . . . . . 15
3635biimpri 197 . . . . . . . . . . . . . 14
3734, 36mpan 651 . . . . . . . . . . . . 13
38 ssid 3210 . . . . . . . . . . . . . . 15
3934tz7.48lem 6469 . . . . . . . . . . . . . . 15
4038, 39mpan 651 . . . . . . . . . . . . . 14
41 fnresdm 5369 . . . . . . . . . . . . . . . . 17
4234, 41ax-mp 8 . . . . . . . . . . . . . . . 16
4342cnveqi 4872 . . . . . . . . . . . . . . 15
4443funeqi 5291 . . . . . . . . . . . . . 14
4540, 44sylib 188 . . . . . . . . . . . . 13
4637, 45anim12i 549 . . . . . . . . . . . 12
47 df-f1 5276 . . . . . . . . . . . 12
4846, 47sylibr 203 . . . . . . . . . . 11
49 f1dmex 5767 . . . . . . . . . . 11
5048, 1, 49sylancl 643 . . . . . . . . . 10
5150ex 423 . . . . . . . . 9
5231, 51syld 40 . . . . . . . 8
5352exp3acom23 1362 . . . . . . 7
5453imp 418 . . . . . 6
5520, 54sylbi 187 . . . . 5
5619, 55mtoi 169 . . . 4
5756rexlimivw 2676 . . 3
58 fveq2 5541 . . . . . 6
59 id 19 . . . . . 6
6058, 59eqeq12d 2310 . . . . 5
6160cbvrexv 2778 . . . 4
62 dfrex2 2569 . . . 4
6361, 62bitr2i 241 . . 3
6457, 63sylib 188 . 2
6518, 64impbii 180 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358   wceq 1632   wcel 1696  wral 2556  wrex 2557  cvv 2801   wss 3165   cmpt 4093  con0 4408  ccnv 4704   cres 4707   wfun 5265   wfn 5266  wf 5267  wf1 5268  cfv 5271  crdg 6438 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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