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Theorem fnxpdmdm 44403
 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 6425 . 2 (𝐹 Fn (𝐴 × 𝐴) → dom 𝐹 = (𝐴 × 𝐴))
2 dmeq 5736 . . 3 (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = dom (𝐴 × 𝐴))
3 dmxpid 5764 . . 3 dom (𝐴 × 𝐴) = 𝐴
42, 3eqtrdi 2849 . 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 1538   × cxp 5517  dom cdm 5519   Fn wfn 6319 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-dm 5529  df-fn 6327 This theorem is referenced by: (None)
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