MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wdomimag Structured version   Visualization version   GIF version

Theorem wdomimag 9044
Description: A set is weakly dominant over its image under any function. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
wdomimag ((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ≼* 𝐴)

Proof of Theorem wdomimag
StepHypRef Expression
1 funimaexg 6429 . 2 ((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ∈ V)
2 wdomima2g 9043 . 2 ((Fun 𝐹𝐴𝑉 ∧ (𝐹𝐴) ∈ V) → (𝐹𝐴) ≼* 𝐴)
31, 2mpd3an3 1459 1 ((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ≼* 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  Vcvv 3480   class class class wbr 5053  cima 5546  Fun wfun 6338  * cwdom 9021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-wdom 9022
This theorem is referenced by:  hsmexlem4  9845
  Copyright terms: Public domain W3C validator