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

Theorem abrexco 6156
Description: Composition of two image maps ( ) and ( ). (Contributed by NM, 27-May-2013.)
Hypotheses
Ref Expression
abrexco.1
abrexco.2
Assertion
Ref Expression
abrexco
Distinct variable groups:   , ,   , ,   ,   ,   , ,   ,

Proof of Theorem abrexco
StepHypRef Expression
1 df-rex 2813 . . . . 5
2 vex 3112 . . . . . . . . 9
3 eqeq1 2461 . . . . . . . . . 10
43rexbidv 2968 . . . . . . . . 9
52, 4elab 3246 . . . . . . . 8
65anbi1i 695 . . . . . . 7
7 r19.41v 3009 . . . . . . 7
86, 7bitr4i 252 . . . . . 6
98exbii 1667 . . . . 5
101, 9bitri 249 . . . 4
11 rexcom4 3129 . . . 4
1210, 11bitr4i 252 . . 3
13 abrexco.1 . . . . 5
14 abrexco.2 . . . . . 6
1514eqeq2d 2471 . . . . 5
1613, 15ceqsexv 3146 . . . 4
1716rexbii 2959 . . 3
1812, 17bitri 249 . 2
1918abbii 2591 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  E.wrex 2808   cvv 3109
This theorem is referenced by:  rankcf  9176  sylow1lem2  16619  sylow3lem1  16647  restco  19665
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-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-v 3111
  Copyright terms: Public domain W3C validator