Theorem dmdm 49528

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 6601	. . 3 ⊢ (𝐴 Fn (𝐵 × 𝐵) → dom 𝐴 = (𝐵 × 𝐵))
2	1	dmeqd 5860	. 2 ⊢ (𝐴 Fn (𝐵 × 𝐵) → dom dom 𝐴 = dom (𝐵 × 𝐵))
3		dmxpid 5885	. 2 ⊢ dom (𝐵 × 𝐵) = 𝐵
4	2, 3	eqtr2di 2788	1 ⊢ (𝐴 Fn (𝐵 × 𝐵) → 𝐵 = dom dom 𝐴)

Syntax hints:   → wi 4   = wceq 1542   × cxp 5629  dom cdm 5631   Fn wfn 6493

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-dm 5641  df-fn 6501

This theorem is referenced by:  iinfconstbas  49541