Theorem fnxpdmdm 48348

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

Step	Hyp	Ref	Expression
1		fndm 6593	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom 𝐹 = (𝐴 × 𝐴))
2		dmeq 5850	. . 3 ⊢ (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = dom (𝐴 × 𝐴))
3		dmxpid 5877	. . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴
4	2, 3	eqtrdi 2785	. 2 ⊢ (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)
5	1, 4	syl 17	1 ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   × cxp 5620  dom cdm 5622   Fn wfn 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-dm 5632  df-fn 6493

This theorem is referenced by: (None)