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Theorem rngmgmbs3 25520
 Description: The domain of the first variable of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.)
Assertion
Ref Expression
rngmgmbs3
Distinct variable group:   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem rngmgmbs3
StepHypRef Expression
1 rexn0 3569 . . 3
2 fdm 5409 . . . 4
3 dmeq 4895 . . . . 5
4 dmxp 4913 . . . . 5
5 eqtr 2313 . . . . . 6
65ex 423 . . . . 5
73, 4, 6syl2im 34 . . . 4
82, 7syl 15 . . 3
91, 8syl5com 26 . 2
109impcom 419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1632   wne 2459  wral 2556  wrex 2557  c0 3468   cxp 4703   cdm 4705  wf 5267  (class class class)co 5874 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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-fn 5274  df-f 5275
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