Theorem rnxpid 5807

Description: The range of a square Cartesian product. (Contributed by FL, 17-May-2010.)

Assertion

Ref	Expression
rnxpid	⊢ ran (𝐴 × 𝐴) = 𝐴

Proof of Theorem rnxpid

Step	Hyp	Ref	Expression
1		rn0 5609	. . 3 ⊢ ran ∅ = ∅
2		xpeq2 5362	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅))
3		xp0 5792	. . . . 5 ⊢ (𝐴 × ∅) = ∅
4	2, 3	syl6eq 2876	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
5	4	rneqd 5584	. . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅)
6		id 22	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
7	1, 5, 6	3eqtr4a 2886	. 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴)
8		rnxp 5804	. 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴)
9	7, 8	pm2.61ine 3081	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   = wceq 1658  ∅c0 4143   × cxp 5339  ran crn 5342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-opab 4935  df-xp 5347  df-rel 5348  df-cnv 5349  df-dm 5351  df-rn 5352

This theorem is referenced by:  sofld  5821  fpwwe2lem13  9778  ustimasn  22401  utopbas  22408  restutop  22410  ovoliunlem1  23667  metideq  30480  poimirlem3  33955  mblfinlem1  33989  rtrclex  38764