Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1dom Structured version   Unicode version

Theorem f1dom 7129
 Description: The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
Hypothesis
Ref Expression
f1dom.1
Assertion
Ref Expression
f1dom

Proof of Theorem f1dom
StepHypRef Expression
1 f1dom.1 . 2
2 f1domg 7127 . 2
31, 2ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wi 4   wcel 1725  cvv 2956   class class class wbr 4212  wf1 5451   cdom 7107 This theorem is referenced by:  dominf  8325  dominfac  8448  lgsqrlem4  21128 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-dom 7111
 Copyright terms: Public domain W3C validator