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

Theorem ello1 13350
 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 5213 . . . . 5
21ineq1d 3695 . . . 4
3 fveq1 5871 . . . . 5
43breq1d 4466 . . . 4
52, 4raleqbidv 3068 . . 3
652rexbidv 2975 . 2
7 df-lo1 13326 . 2
86, 7elrab2 3259 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   wceq 1395   wcel 1819  wral 2807  wrex 2808   cin 3470   class class class wbr 4456   cdm 5008  cfv 5594  (class class class)co 6296   cpm 7439  cr 9508   cpnf 9642   cle 9646  cico 11556  clo1 13322 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  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 1614  df-nf 1618  df-sb 1741  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 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-dm 5018  df-iota 5557  df-fv 5602  df-lo1 13326 This theorem is referenced by:  ello12  13351  lo1f  13353  lo1dm  13354
 Copyright terms: Public domain W3C validator