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Theorem xptrrel 14328
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 4202 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵)
2 dmxpss 6021 . . . . . . . 8 dom (𝐴 × 𝐵) ⊆ 𝐴
31, 2sstri 3973 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴
4 inss2 4203 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
5 rnxpss 6022 . . . . . . . 8 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5sstri 3973 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵
73, 6ssini 4205 . . . . . 6 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴𝐵)
8 eqimss 4020 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐴𝐵) ⊆ ∅)
97, 8sstrid 3975 . . . . 5 ((𝐴𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅)
10 ss0 4349 . . . . 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 14327 . . 3 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅)
13 0ss 4347 . . 3 ∅ ⊆ (𝐴 × 𝐵)
1412, 13eqsstrdi 4018 . 2 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
15 neqne 3021 . . . 4 (¬ (𝐴𝐵) = ∅ → (𝐴𝐵) ≠ ∅)
1615xpcoidgend 14323 . . 3 (¬ (𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
17 ssid 3986 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵)
1816, 17eqsstrdi 4018 . 2 (¬ (𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
1914, 18pm2.61i 183 1 ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1528  cin 3932  wss 3933  c0 4288   × cxp 5546  dom cdm 5548  ran crn 5549  ccom 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560
This theorem is referenced by:  trclublem  14343  trclubgNEW  39856  trclexi  39858  cnvtrcl0  39864  xpintrreld  39889  trrelsuperreldg  39891  trrelsuperrel2dg  39894
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