Theorem foima 6794

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 6066	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 6789	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6713	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
4	3	imaeq2d 6056	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5		forn 6792	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6	1, 4, 5	3eqtr3a 2819	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  dom cdm 5655  ran crn 5656   “ cima 5658  –onto→wfo 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fn 6536  df-f 6537  df-fo 6539

This theorem is used by:  foimacnv  6835  fodomfi  9282  domunfican  9291  fiint  9296  cantnflt2  9652  cantnfp1lem3  9659  enfin1ai  10386  symgfixelsi  19562  dprdf1o  20161  lmimlbs  22049  cncmp  23617  cmpfi  23633  cnconn  23647  qtopval2  23922  elfm3  24176  rnelfm  24179  fmfnfmlem2  24181  fmfnfm  24184  eupthvdres  30715  pjordi  32654  qtophaus  34346  poimirlem1  38370  poimirlem2  38371  poimirlem3  38372  poimirlem4  38373  poimirlem5  38374  poimirlem6  38375  poimirlem7  38376  poimirlem9  38378  poimirlem10  38379  poimirlem11  38380  poimirlem12  38381  poimirlem14  38383  poimirlem16  38385  poimirlem17  38386  poimirlem19  38388  poimirlem20  38389  poimirlem22  38391  poimirlem23  38392  poimirlem24  38393  poimirlem25  38394  poimirlem29  38398  poimirlem31  38400  ovoliunnfl  38411  voliunnfl  38413  volsupnfl  38414  ismtybndlem  38556  riccrng1  43403  ricdrng1  43410  kelac1  43904  gicabl  43940  imasubc  50077