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Theorem xpcoid 6311
Description: Composition of two Cartesian squares. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
xpcoid ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)

Proof of Theorem xpcoid
StepHypRef Expression
1 co01 6282 . . 3 (∅ ∘ ∅) = ∅
2 id 22 . . . . . 6 (𝐴 = ∅ → 𝐴 = ∅)
32sqxpeqd 5720 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × ∅))
4 0xp 5786 . . . . 5 (∅ × ∅) = ∅
53, 4eqtrdi 2790 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
65, 5coeq12d 5877 . . 3 (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (∅ ∘ ∅))
71, 6, 53eqtr4a 2800 . 2 (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
8 xpco 6310 . 2 (𝐴 ≠ ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
97, 8pm2.61ine 3022 1 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  c0 4338   × cxp 5686  ccom 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697
This theorem is referenced by:  utop2nei  24274
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