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Theorem fnrndomg 8176
Description: The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)
Assertion
Ref Expression
fnrndomg  |-  ( A  e.  B  ->  ( F  Fn  A  ->  ran 
F  ~<_  A ) )

Proof of Theorem fnrndomg
StepHypRef Expression
1 dffn4 5473 . 2  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
2 fodomg 8166 . 2  |-  ( A  e.  B  ->  ( F : A -onto-> ran  F  ->  ran  F  ~<_  A ) )
31, 2syl5bi 208 1  |-  ( A  e.  B  ->  ( F  Fn  A  ->  ran 
F  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   class class class wbr 4039   ran crn 4706    Fn wfn 5266   -onto->wfo 5269    ~<_ cdom 6877
This theorem is referenced by:  unirnfdomd  8205  konigthlem  8206  abrexdomjm  23181  fnct  23356  abrexdom  26508  indexdom  26516
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-ac2 8105
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-card 7588  df-acn 7591  df-ac 7759
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