Theorem dmdm 49164

Description: The double domain of a function on a Cartesian square. (Contributed by Zhi Wang, 1-Nov-2025.)

Assertion

Ref	Expression
dmdm	⊢ (𝐴 Fn (𝐵 × 𝐵) → 𝐵 = dom dom 𝐴)

Proof of Theorem dmdm

Step	Hyp	Ref	Expression
1		fndm 6584	. . 3 ⊢ (𝐴 Fn (𝐵 × 𝐵) → dom 𝐴 = (𝐵 × 𝐵))
2	1	dmeqd 5844	. 2 ⊢ (𝐴 Fn (𝐵 × 𝐵) → dom dom 𝐴 = dom (𝐵 × 𝐵))
3		dmxpid 5869	. 2 ⊢ dom (𝐵 × 𝐵) = 𝐵
4	2, 3	eqtr2di 2783	1 ⊢ (𝐴 Fn (𝐵 × 𝐵) → 𝐵 = dom dom 𝐴)

Syntax hints:   → wi 4   = wceq 1541   × cxp 5612  dom cdm 5614   Fn wfn 6476

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-dm 5624  df-fn 6484

This theorem is referenced by:  iinfconstbas  49177