Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnxpdmdm Structured version   Visualization version   GIF version

Theorem fnxpdmdm 47300
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
StepHypRef Expression
1 fndm 6662 . 2 (𝐹 Fn (𝐴 × 𝐴) → dom 𝐹 = (𝐴 × 𝐴))
2 dmeq 5910 . . 3 (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = dom (𝐴 × 𝐴))
3 dmxpid 5936 . . 3 dom (𝐴 × 𝐴) = 𝐴
42, 3eqtrdi 2784 . 2 (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)
51, 4syl 17 1 (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   × cxp 5680  dom cdm 5682   Fn wfn 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-dm 5692  df-fn 6556
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator