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Theorem fodmrnu 5277
 Description: An onto function has unique domain and range. (Contributed by set.mm contributors, 5-Nov-2006.)
Assertion
Ref Expression
fodmrnu ((F:AontoB F:ContoD) → (A = C B = D))

Proof of Theorem fodmrnu
StepHypRef Expression
1 fofn 5271 . . 3 (F:AontoBF Fn A)
2 fofn 5271 . . 3 (F:ContoDF Fn C)
3 fndmu 5184 . . 3 ((F Fn A F Fn C) → A = C)
41, 2, 3syl2an 463 . 2 ((F:AontoB F:ContoD) → A = C)
5 forn 5272 . . 3 (F:AontoB → ran F = B)
6 forn 5272 . . 3 (F:ContoD → ran F = D)
75, 6sylan9req 2406 . 2 ((F:AontoB F:ContoD) → B = D)
84, 7jca 518 1 ((F:AontoB F:ContoD) → (A = C B = D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ran crn 4773   Fn wfn 4776  –onto→wfo 4779 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-fn 4790  df-f 4791  df-fo 4793 This theorem is referenced by: (None)
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