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Theorem fodmrnu 6581
 Description: An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
Assertion
Ref Expression
fodmrnu ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem fodmrnu
StepHypRef Expression
1 fofn 6575 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 fofn 6575 . . 3 (𝐹:𝐶onto𝐷𝐹 Fn 𝐶)
3 fndmu 6437 . . 3 ((𝐹 Fn 𝐴𝐹 Fn 𝐶) → 𝐴 = 𝐶)
41, 2, 3syl2an 598 . 2 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → 𝐴 = 𝐶)
5 forn 6576 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
6 forn 6576 . . 3 (𝐹:𝐶onto𝐷 → ran 𝐹 = 𝐷)
75, 6sylan9req 2854 . 2 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → 𝐵 = 𝐷)
84, 7jca 515 1 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ran crn 5524   Fn wfn 6327  –onto→wfo 6330 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-in 3890  df-ss 3900  df-fn 6335  df-f 6336  df-fo 6338 This theorem is referenced by: (None)
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