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Theorem fnxpdmdm 41558
 Description: The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.)
Assertion
Ref Expression
fnxpdmdm (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)

Proof of Theorem fnxpdmdm
StepHypRef Expression
1 fndm 5904 . 2 (𝐹 Fn (𝐴 × 𝐴) → dom 𝐹 = (𝐴 × 𝐴))
2 dmeq 5246 . . 3 (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = dom (𝐴 × 𝐴))
3 dmxpid 5266 . . 3 dom (𝐴 × 𝐴) = 𝐴
42, 3syl6eq 2660 . 2 (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)
51, 4syl 17 1 (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   × cxp 5036  dom cdm 5038   Fn wfn 5799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-fn 5807 This theorem is referenced by: (None)
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