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Theorem fof 5031
Description: An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
fof (𝐹:AontoB𝐹:AB)

Proof of Theorem fof
StepHypRef Expression
1 eqimss 2974 . . 3 (ran 𝐹 = B → ran 𝐹B)
21anim2i 324 . 2 ((𝐹 Fn A ran 𝐹 = B) → (𝐹 Fn A ran 𝐹B))
3 df-fo 4835 . 2 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
4 df-f 4833 . 2 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
52, 3, 43imtr4i 190 1 (𝐹:AontoB𝐹:AB)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wss 2894  ran crn 4273   Fn wfn 4824  wf 4825  ontowfo 4827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908  df-f 4833  df-fo 4835
This theorem is referenced by:  fofun  5032  fofn  5033  dffo2  5035  foima  5036  resdif  5073  ffoss  5083  fconstfvm  5304  cocan2  5353  foeqcnvco  5355  fornex  5665  algrflem  5773  tposf2  5805
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