Theorem foima 6597

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 5913	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6592	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6515	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 5903	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6595	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2797	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1542  dom cdm 5525  ran crn 5526   “ cima 5528  –onto→wfo 6337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-fn 6342  df-f 6343  df-fo 6345

This theorem is referenced by:  foimacnv  6637  domunfican  8867  fiint  8871  fodomfi  8872  cantnflt2  9211  cantnfp1lem3  9218  enfin1ai  9886  symgfixelsi  18683  dprdf1o  19275  lmimlbs  20654  cncmp  22145  cmpfi  22161  cnconn  22175  qtopval2  22449  elfm3  22703  rnelfm  22706  fmfnfmlem2  22708  fmfnfm  22711  eupthvdres  28174  pjordi  30110  qtophaus  31360  poimirlem1  35423  poimirlem2  35424  poimirlem3  35425  poimirlem4  35426  poimirlem5  35427  poimirlem6  35428  poimirlem7  35429  poimirlem9  35431  poimirlem10  35432  poimirlem11  35433  poimirlem12  35434  poimirlem14  35436  poimirlem16  35438  poimirlem17  35439  poimirlem19  35441  poimirlem20  35442  poimirlem22  35444  poimirlem23  35445  poimirlem24  35446  poimirlem25  35447  poimirlem29  35451  poimirlem31  35453  ovoliunnfl  35464  voliunnfl  35466  volsupnfl  35467  ismtybndlem  35609  kelac1  40482  gicabl  40518