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Theorem xptrrel 14925
Description: The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.)
Assertion
Ref Expression
xptrrel ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)

Proof of Theorem xptrrel
StepHypRef Expression
1 inss1 4221 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵)
2 dmxpss 6161 . . . . . . . 8 dom (𝐴 × 𝐵) ⊆ 𝐴
31, 2sstri 3984 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴
4 inss2 4222 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
5 rnxpss 6162 . . . . . . . 8 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5sstri 3984 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵
73, 6ssini 4224 . . . . . 6 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴𝐵)
8 eqimss 4033 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐴𝐵) ⊆ ∅)
97, 8sstrid 3986 . . . . 5 ((𝐴𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅)
10 ss0 4391 . . . . 5 ((dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) = ∅)
119, 10syl 17 . . . 4 ((𝐴𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) = ∅)
1211coemptyd 14924 . . 3 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅)
13 0ss 4389 . . 3 ∅ ⊆ (𝐴 × 𝐵)
1412, 13eqsstrdi 4029 . 2 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
15 neqne 2940 . . . 4 (¬ (𝐴𝐵) = ∅ → (𝐴𝐵) ≠ ∅)
1615xpcoidgend 14920 . . 3 (¬ (𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
17 ssid 3997 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵)
1816, 17eqsstrdi 4029 . 2 (¬ (𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
1914, 18pm2.61i 182 1 ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  cin 3940  wss 3941  c0 4315   × cxp 5665  dom cdm 5667  ran crn 5668  ccom 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679
This theorem is referenced by:  trclublem  14940  trclubgNEW  42883  trclexi  42885  cnvtrcl0  42891  xpintrreld  42931  trrelsuperreldg  42933  trrelsuperrel2dg  42936
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