Theorem rnxpid 6030

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 5796	. . 3 ⊢ ran ∅ = ∅
2		xpeq2 5576	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅))
3		xp0 6015	. . . . 5 ⊢ (𝐴 × ∅) = ∅
4	2, 3	syl6eq 2872	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
5	4	rneqd 5808	. . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅)
6		id 22	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
7	1, 5, 6	3eqtr4a 2882	. 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴)
8		rnxp 6027	. 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴)
9	7, 8	pm2.61ine 3100	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   = wceq 1537  ∅c0 4291   × cxp 5553  ran crn 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563  df-dm 5565  df-rn 5566

This theorem is referenced by:  sofld  6044  fpwwe2lem13  10064  ustimasn  22837  utopbas  22844  restutop  22846  ovoliunlem1  24103  metideq  31133  poimirlem3  34910  mblfinlem1  34944  rtrclex  40026