Theorem foima 6795

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

Step	Hyp	Ref	Expression
1		imadmrn 6070	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6790	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6714	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 6060	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6793	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2828	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1567  dom cdm 5659  ran crn 5660   “ cima 5662  –onto→wfo 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fn 6536  df-f 6537  df-fo 6539

This theorem is referenced by:  foimacnv  6836  fodomfi  9268  domunfican  9277  fiint  9282  cantnflt2  9638  cantnfp1lem3  9645  enfin1ai  10364  symgfixelsi  19501  dprdf1o  20100  lmimlbs  21951  cncmp  23514  cmpfi  23530  cnconn  23544  qtopval2  23818  elfm3  24072  rnelfm  24075  fmfnfmlem2  24077  fmfnfm  24080  eupthvdres  30523  pjordi  32462  qtophaus  34167  poimirlem1  38155  poimirlem2  38156  poimirlem3  38157  poimirlem4  38158  poimirlem5  38159  poimirlem6  38160  poimirlem7  38161  poimirlem9  38163  poimirlem10  38164  poimirlem11  38165  poimirlem12  38166  poimirlem14  38168  poimirlem16  38170  poimirlem17  38171  poimirlem19  38173  poimirlem20  38174  poimirlem22  38176  poimirlem23  38177  poimirlem24  38178  poimirlem25  38179  poimirlem29  38183  poimirlem31  38185  ovoliunnfl  38196  voliunnfl  38198  volsupnfl  38199  ismtybndlem  38340  riccrng1  43174  ricdrng1  43181  kelac1  43675  gicabl  43711  imasubc  49807