Theorem fnxpdmdm 47194

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

Step	Hyp	Ref	Expression
1		fndm 6651	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom 𝐹 = (𝐴 × 𝐴))
2		dmeq 5900	. . 3 ⊢ (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = dom (𝐴 × 𝐴))
3		dmxpid 5926	. . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴
4	2, 3	eqtrdi 2783	. 2 ⊢ (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)
5	1, 4	syl 17	1 ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)

Syntax hints:   → wi 4   = wceq 1534   × cxp 5670  dom cdm 5672   Fn wfn 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-dm 5682  df-fn 6545

This theorem is referenced by: (None)