MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xptrrel Structured version   Visualization version   GIF version

Theorem xptrrel 15015
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 4244 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵)
2 dmxpss 6192 . . . . . . . 8 dom (𝐴 × 𝐵) ⊆ 𝐴
31, 2sstri 4004 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴
4 inss2 4245 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
5 rnxpss 6193 . . . . . . . 8 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5sstri 4004 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵
73, 6ssini 4247 . . . . . 6 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴𝐵)
8 eqimss 4053 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐴𝐵) ⊆ ∅)
97, 8sstrid 4006 . . . . 5 ((𝐴𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅)
10 ss0 4407 . . . . 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 15014 . . 3 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅)
13 0ss 4405 . . 3 ∅ ⊆ (𝐴 × 𝐵)
1412, 13eqsstrdi 4049 . 2 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
15 neqne 2945 . . . 4 (¬ (𝐴𝐵) = ∅ → (𝐴𝐵) ≠ ∅)
1615xpcoidgend 15010 . . 3 (¬ (𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
17 ssid 4017 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵)
1816, 17eqsstrdi 4049 . 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 1536  cin 3961  wss 3962  c0 4338   × cxp 5686  dom cdm 5688  ran crn 5689  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-10 2138  ax-11 2154  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-nf 1780  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-in 3969  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  df-dm 5698  df-rn 5699  df-res 5700
This theorem is referenced by:  trclublem  15030  trclubgNEW  43607  trclexi  43609  cnvtrcl0  43615  xpintrreld  43655  trrelsuperreldg  43657  trrelsuperrel2dg  43660
  Copyright terms: Public domain W3C validator