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Theorem foima 5473
Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
Assertion
Ref Expression
foima (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)

Proof of Theorem foima
StepHypRef Expression
1 imadmrn 5009 . 2 (𝐹 “ dom 𝐹) = ran 𝐹
2 fof 5468 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
3 fdm 5401 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
42, 3syl 14 . . 3 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
54imaeq2d 4999 . 2 (𝐹:𝐴onto𝐵 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
6 forn 5471 . 2 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
71, 5, 63eqtr3a 2250 1 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  dom cdm 4655  ran crn 4656  cima 4658  wf 5242  ontowfo 5244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fn 5249  df-f 5250  df-fo 5252
This theorem is referenced by:  foimacnv  5510  foima2  5786  fiintim  6976  fidcenumlemr  7004
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