Theorem rnxpid 6163

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 5907	. . 3 ⊢ ran ∅ = ∅
2		xpeq2 5673	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅))
3		xp0 5752	. . . . 5 ⊢ (𝐴 × ∅) = ∅
4	2, 3	eqtrdi 2816	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
5	4	rneqd 5919	. . 3 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅)
6		id 23	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
7	1, 5, 6	3eqtr4a 2826	. 2 ⊢ (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴)
8		rnxp 6160	. 2 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴)
9	7, 8	pm2.61ine 3043	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   = wceq 1563  ∅c0 4288   × cxp 5650  ran crn 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663

This theorem is referenced by:  sofld  6177  fpwwe2lem12  10615  ustimasn  24346  utopbas  24353  restutop  24355  ovoliunlem1  25622  metideq  34200  poimirlem3  38134  mblfinlem1  38168  rtrclex  44205