Theorem dmdm 48914

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 6638	. . 3 ⊢ (𝐴 Fn (𝐵 × 𝐵) → dom 𝐴 = (𝐵 × 𝐵))
2	1	dmeqd 5883	. 2 ⊢ (𝐴 Fn (𝐵 × 𝐵) → dom dom 𝐴 = dom (𝐵 × 𝐵))
3		dmxpid 5908	. 2 ⊢ dom (𝐵 × 𝐵) = 𝐵
4	2, 3	eqtr2di 2786	1 ⊢ (𝐴 Fn (𝐵 × 𝐵) → 𝐵 = dom dom 𝐴)

Syntax hints:   → wi 4   = wceq 1539   × cxp 5650  dom cdm 5652   Fn wfn 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pr 5400

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-br 5118  df-opab 5180  df-xp 5658  df-dm 5662  df-fn 6531

This theorem is referenced by:  iinfconstbas  48927