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Theorem rnxpid 6129
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
StepHypRef Expression
1 rn0 5885 . . 3 ran ∅ = ∅
2 xpeq2 5658 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅))
3 xp0 6114 . . . . 5 (𝐴 × ∅) = ∅
42, 3eqtrdi 2789 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
54rneqd 5897 . . 3 (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅)
6 id 22 . . 3 (𝐴 = ∅ → 𝐴 = ∅)
71, 5, 63eqtr4a 2799 . 2 (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴)
8 rnxp 6126 . 2 (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴)
97, 8pm2.61ine 3025 1 ran (𝐴 × 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  c0 4286   × cxp 5635  ran crn 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648
This theorem is referenced by:  sofld  6143  fpwwe2lem12  10586  ustimasn  23603  utopbas  23610  restutop  23612  ovoliunlem1  24889  metideq  32538  poimirlem3  36131  mblfinlem1  36165  rtrclex  41981
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