Theorem rnxpid 6130

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

Assertion

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

Proof of Theorem rnxpid

Step	Hyp	Ref	Expression
1		rn0 5886	. . 3 ⊢ ran ∅ = ∅
2		xpeq2 5659	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅))
3		xp0 6115	. . . . 5 ⊢ (𝐴 × ∅) = ∅
4	2, 3	eqtrdi 2787	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
5	4	rneqd 5898	. . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅)
6		id 22	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
7	1, 5, 6	3eqtr4a 2797	. 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴)
8		rnxp 6127	. 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴)
9	7, 8	pm2.61ine 3024	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   = wceq 1541  ∅c0 4287   × cxp 5636  ran crn 5639

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649

This theorem is referenced by:  sofld  6144  fpwwe2lem12  10587  ustimasn  23617  utopbas  23624  restutop  23626  ovoliunlem1  24903  metideq  32563  poimirlem3  36154  mblfinlem1  36188  rtrclex  42011