MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ello1 Unicode version

Theorem ello1 13338
Description: Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
Assertion
Ref Expression
ello1
Distinct variable group:   , , ,

Proof of Theorem ello1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dmeq 5208 . . . . 5
21ineq1d 3698 . . . 4
3 fveq1 5870 . . . . 5
43breq1d 4462 . . . 4
52, 4raleqbidv 3068 . . 3
652rexbidv 2975 . 2
7 df-lo1 13314 . 2
86, 7elrab2 3259 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  i^icin 3474   class class class wbr 4452  domcdm 5004  `cfv 5593  (class class class)co 6296   cpm 7440   cr 9512   cpnf 9646   cle 9650   cico 11560   clo1 13310
This theorem is referenced by:  ello12  13339  lo1f  13341  lo1dm  13342
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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  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-dm 5014  df-iota 5556  df-fv 5601  df-lo1 13314
  Copyright terms: Public domain W3C validator