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Theorem xpcoid 5837
Description: Composition of two square Cartesian products. (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 5811 . . 3 (∅ ∘ ∅) = ∅
2 id 22 . . . . . 6 (𝐴 = ∅ → 𝐴 = ∅)
32sqxpeqd 5298 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × ∅))
4 0xp 5356 . . . . 5 (∅ × ∅) = ∅
53, 4syl6eq 2810 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
65, 5coeq12d 5442 . . 3 (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (∅ ∘ ∅))
71, 6, 53eqtr4a 2820 . 2 (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
8 xpco 5836 . 2 (𝐴 ≠ ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
97, 8pm2.61ine 3015 1 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  c0 4058   × cxp 5264  ccom 5270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275
This theorem is referenced by:  utop2nei  22255
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