Theorem dmdm 49890

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 6642	. . 3 ⊢ (𝐴 Fn (𝐵 × 𝐵) → dom 𝐴 = (𝐵 × 𝐵))
2	1	dmeqd 5897	. 2 ⊢ (𝐴 Fn (𝐵 × 𝐵) → dom dom 𝐴 = dom (𝐵 × 𝐵))
3		dmxpid 5922	. 2 ⊢ dom (𝐵 × 𝐵) = 𝐵
4	2, 3	eqtr2di 2817	1 ⊢ (𝐴 Fn (𝐵 × 𝐵) → 𝐵 = dom dom 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   × cxp 5661  dom cdm 5663   Fn wfn 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-dm 5673  df-fn 6543

This theorem is used by:  iinfconstbas  49903