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Theorem xptrrel 14874
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 4192 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵)
2 dmxpss 6127 . . . . . . . 8 dom (𝐴 × 𝐵) ⊆ 𝐴
31, 2sstri 3957 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴
4 inss2 4193 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
5 rnxpss 6128 . . . . . . . 8 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5sstri 3957 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵
73, 6ssini 4195 . . . . . 6 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴𝐵)
8 eqimss 4004 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐴𝐵) ⊆ ∅)
97, 8sstrid 3959 . . . . 5 ((𝐴𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅)
10 ss0 4362 . . . . 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 14873 . . 3 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅)
13 0ss 4360 . . 3 ∅ ⊆ (𝐴 × 𝐵)
1412, 13eqsstrdi 4002 . 2 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
15 neqne 2948 . . . 4 (¬ (𝐴𝐵) = ∅ → (𝐴𝐵) ≠ ∅)
1615xpcoidgend 14869 . . 3 (¬ (𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
17 ssid 3970 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵)
1816, 17eqsstrdi 4002 . 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 1542  cin 3913  wss 3914  c0 4286   × cxp 5635  dom cdm 5637  ran crn 5638  ccom 5641
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-rex 3071  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-co 5646  df-dm 5647  df-rn 5648  df-res 5649
This theorem is referenced by:  trclublem  14889  trclubgNEW  41982  trclexi  41984  cnvtrcl0  41990  xpintrreld  42030  trrelsuperreldg  42032  trrelsuperrel2dg  42035
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