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Theorem fodmrnu 6641
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 6635 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 fofn 6635 . . 3 (𝐹:𝐶onto𝐷𝐹 Fn 𝐶)
3 fndmu 6485 . . 3 ((𝐹 Fn 𝐴𝐹 Fn 𝐶) → 𝐴 = 𝐶)
41, 2, 3syl2an 599 . 2 ((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → 𝐴 = 𝐶)
5 forn 6636 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
6 forn 6636 . . 3 (𝐹:𝐶onto𝐷 → ran 𝐹 = 𝐷)
75, 6sylan9req 2799 . 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 1543  ran crn 5552   Fn wfn 6375  ontowfo 6378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-in 3873  df-ss 3883  df-fn 6383  df-f 6384  df-fo 6386
This theorem is referenced by: (None)
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