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Theorem fodmrnu 6339
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 6333 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 fofn 6333 . . 3 (𝐹:𝐶onto𝐷𝐹 Fn 𝐶)
3 fndmu 6203 . . 3 ((𝐹 Fn 𝐴𝐹 Fn 𝐶) → 𝐴 = 𝐶)
41, 2, 3syl2an 590 . 2 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → 𝐴 = 𝐶)
5 forn 6334 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
6 forn 6334 . . 3 (𝐹:𝐶onto𝐷 → ran 𝐹 = 𝐷)
75, 6sylan9req 2854 . 2 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → 𝐵 = 𝐷)
84, 7jca 508 1 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  ran crn 5313   Fn wfn 6096  ontowfo 6099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-in 3776  df-ss 3783  df-fn 6104  df-f 6105  df-fo 6107
This theorem is referenced by: (None)
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