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

Theorem foima 5660
Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
Assertion
Ref Expression
foima  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )

Proof of Theorem foima
StepHypRef Expression
1 imadmrn 5217 . 2  |-  ( F
" dom  F )  =  ran  F
2 fof 5655 . . . 4  |-  ( F : A -onto-> B  ->  F : A --> B )
3 fdm 5597 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
42, 3syl 16 . . 3  |-  ( F : A -onto-> B  ->  dom  F  =  A )
54imaeq2d 5205 . 2  |-  ( F : A -onto-> B  -> 
( F " dom  F )  =  ( F
" A ) )
6 forn 5658 . 2  |-  ( F : A -onto-> B  ->  ran  F  =  B )
71, 5, 63eqtr3a 2494 1  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   dom cdm 4880   ran crn 4881   "cima 4883   -->wf 5452   -onto->wfo 5454
This theorem is referenced by:  foimacnv  5694  domunfican  7381  fiint  7385  fodomfi  7387  cantnflt2  7630  cantnfp1lem3  7638  enfin1ai  8266  dprdf1o  15592  cncmp  17457  cmpfi  17473  cnconn  17487  qtopval2  17730  elfm3  17984  rnelfm  17987  fmfnfmlem2  17989  fmfnfm  17992  eupath2  21704  pjordi  23678  ovoliunnfl  26250  voliunnfl  26252  volsupnfl  26253  ismtybndlem  26517  kelac1  27140  gicabl  27242  lmimlbs  27285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fn 5459  df-f 5460  df-fo 5462
  Copyright terms: Public domain W3C validator