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Theorem fodmrnu 6829
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 6823 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 fofn 6823 . . 3 (𝐹:𝐶onto𝐷𝐹 Fn 𝐶)
3 fndmu 6676 . . 3 ((𝐹 Fn 𝐴𝐹 Fn 𝐶) → 𝐴 = 𝐶)
41, 2, 3syl2an 596 . 2 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → 𝐴 = 𝐶)
5 forn 6824 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
6 forn 6824 . . 3 (𝐹:𝐶onto𝐷 → ran 𝐹 = 𝐷)
75, 6sylan9req 2796 . 2 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → 𝐵 = 𝐷)
84, 7jca 511 1 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  ran crn 5690   Fn wfn 6558  ontowfo 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ss 3980  df-fn 6566  df-f 6567  df-fo 6569
This theorem is referenced by: (None)
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