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

Theorem abrexco 5782
 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:   ,,   ,,   ,   ,   ,,   ,
Allowed substitution hints:   (,)   (,)   (,,)   (,,)

Proof of Theorem abrexco
StepHypRef Expression
1 df-rex 2562 . . . . 5
2 vex 2804 . . . . . . . . 9
3 eqeq1 2302 . . . . . . . . . 10
43rexbidv 2577 . . . . . . . . 9
52, 4elab 2927 . . . . . . . 8
65anbi1i 676 . . . . . . 7
7 r19.41v 2706 . . . . . . 7
86, 7bitr4i 243 . . . . . 6
98exbii 1572 . . . . 5
101, 9bitri 240 . . . 4
11 rexcom4 2820 . . . 4
1210, 11bitr4i 243 . . 3
13 abrexco.1 . . . . 5
14 abrexco.2 . . . . . 6
1514eqeq2d 2307 . . . . 5
1613, 15ceqsexv 2836 . . . 4
1716rexbii 2581 . . 3
1812, 17bitri 240 . 2
1918abbii 2408 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wex 1531   wceq 1632   wcel 1696  cab 2282  wrex 2557  cvv 2801 This theorem is referenced by:  rankcf  8415  sylow1lem2  14926  sylow3lem1  14954  restco  16911 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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803
 Copyright terms: Public domain W3C validator