Theorem rnxpid 6204

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 5950	. . 3 ⊢ ran ∅ = ∅
2		xpeq2 5721	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅))
3		xp0 6189	. . . . 5 ⊢ (𝐴 × ∅) = ∅
4	2, 3	eqtrdi 2796	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
5	4	rneqd 5963	. . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅)
6		id 22	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
7	1, 5, 6	3eqtr4a 2806	. 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴)
8		rnxp 6201	. 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴)
9	7, 8	pm2.61ine 3031	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   = wceq 1537  ∅c0 4352   × cxp 5698  ran crn 5701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711

This theorem is referenced by:  sofld  6218  fpwwe2lem12  10711  ustimasn  24258  utopbas  24265  restutop  24267  ovoliunlem1  25556  metideq  33839  poimirlem3  37583  mblfinlem1  37617  rtrclex  43579