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

Theorem xptrrel 13661
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 3816 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵)
2 dmxpss 5529 . . . . . . . 8 dom (𝐴 × 𝐵) ⊆ 𝐴
31, 2sstri 3596 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴
4 inss2 3817 . . . . . . . 8 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
5 rnxpss 5530 . . . . . . . 8 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5sstri 3596 . . . . . . 7 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵
73, 6ssini 3819 . . . . . 6 (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴𝐵)
8 eqimss 3641 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐴𝐵) ⊆ ∅)
97, 8syl5ss 3598 . . . . 5 ((𝐴𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅)
10 ss0 3951 . . . . 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 13660 . . 3 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅)
13 0ss 3949 . . 3 ∅ ⊆ (𝐴 × 𝐵)
1412, 13syl6eqss 3639 . 2 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
15 df-ne 2791 . . . . 5 ((𝐴𝐵) ≠ ∅ ↔ ¬ (𝐴𝐵) = ∅)
1615biimpri 218 . . . 4 (¬ (𝐴𝐵) = ∅ → (𝐴𝐵) ≠ ∅)
1716xpcoidgend 13656 . . 3 (¬ (𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
18 ssid 3608 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵)
1917, 18syl6eqss 3639 . 2 (¬ (𝐴𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
2014, 19pm2.61i 176 1 ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wne 2790  cin 3558  wss 3559  c0 3896   × cxp 5077  dom cdm 5079  ran crn 5080  ccom 5083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091
This theorem is referenced by:  trclublem  13676  trclubgNEW  37441  trclexi  37443  cnvtrcl0  37449  xpintrreld  37474  trrelsuperreldg  37476  trrelsuperrel2dg  37479
  Copyright terms: Public domain W3C validator