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Theorem fodmrnu 6597
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 6591 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 fofn 6591 . . 3 (𝐹:𝐶onto𝐷𝐹 Fn 𝐶)
3 fndmu 6457 . . 3 ((𝐹 Fn 𝐴𝐹 Fn 𝐶) → 𝐴 = 𝐶)
41, 2, 3syl2an 595 . 2 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → 𝐴 = 𝐶)
5 forn 6592 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
6 forn 6592 . . 3 (𝐹:𝐶onto𝐷 → ran 𝐹 = 𝐷)
75, 6sylan9req 2882 . 2 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → 𝐵 = 𝐷)
84, 7jca 512 1 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  ran crn 5555   Fn wfn 6349  ontowfo 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-in 3947  df-ss 3956  df-fn 6357  df-f 6358  df-fo 6360
This theorem is referenced by: (None)
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