Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  smoiso Unicode version

Theorem smoiso 7052
 Description: If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
Assertion
Ref Expression
smoiso

Proof of Theorem smoiso
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 6221 . . . 4
2 f1of 5821 . . . 4
31, 2syl 16 . . 3
4 ffdm 5750 . . . . . 6
54simpld 459 . . . . 5
6 fss 5744 . . . . 5
75, 6sylan 471 . . . 4
873adant2 1015 . . 3
93, 8syl3an1 1261 . 2
10 fdm 5740 . . . . . 6
1110eqcomd 2465 . . . . 5
12 ordeq 4890 . . . . 5
131, 2, 11, 124syl 21 . . . 4
1413biimpa 484 . . 3
15143adant3 1016 . 2
1610eleq2d 2527 . . . . . . 7
1710eleq2d 2527 . . . . . . 7
1816, 17anbi12d 710 . . . . . 6
191, 2, 183syl 20 . . . . 5
20 isorel 6222 . . . . . . . 8
21 epel 4799 . . . . . . . 8
22 fvex 5881 . . . . . . . . 9
2322epelc 4798 . . . . . . . 8
2420, 21, 233bitr3g 287 . . . . . . 7
2524biimpd 207 . . . . . 6
2625ex 434 . . . . 5
2719, 26sylbid 215 . . . 4
2827ralrimivv 2877 . . 3
29283ad2ant1 1017 . 2
30 df-smo 7036 . 2
319, 15, 29, 30syl3anbrc 1180 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  C_wss 3475   class class class wbr 4452   cep 4794  Ordword 4882   con0 4883  domcdm 5004  -->wf 5589  -1-1-onto->wf1o 5592  cfv 5593  Isomwiso 5594  Smo`wsmo 7035 This theorem is referenced by:  smoiso2  7059 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-iota 5556  df-fn 5596  df-f 5597  df-f1 5598  df-f1o 5600  df-fv 5601  df-isom 5602  df-smo 7036
 Copyright terms: Public domain W3C validator