Theorem xptrrel 14897

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

Step	Hyp	Ref	Expression
1		inss1 4188	. . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵)
2		dmxpss 6126	. . . . . . . 8 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
3	1, 2	sstri 3941	. . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴
4		inss2 4189	. . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
5		rnxpss 6127	. . . . . . . 8 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
6	4, 5	sstri 3941	. . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵
7	3, 6	ssini 4191	. . . . . 6 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴 ∩ 𝐵)
8		eqimss 3990	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ⊆ ∅)
9	7, 8	sstrid 3943	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅)
10		ss0 4353	. . . . 5 ⊢ ((dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) = ∅)
11	9, 10	syl 17	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) = ∅)
12	11	coemptyd 14896	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅)
13		0ss 4351	. . 3 ⊢ ∅ ⊆ (𝐴 × 𝐵)
14	12, 13	eqsstrdi 3976	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
15		neqne 2938	. . . 4 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ≠ ∅)
16	15	xpcoidgend 14892	. . 3 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
17		ssid 3954	. . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵)
18	16, 17	eqsstrdi 3976	. 2 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
19	14, 18	pm2.61i 182	1 ⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)

Syntax hints:  ¬ wn 3   = wceq 1541   ∩ cin 3898   ⊆ wss 3899  ∅c0 4284   × cxp 5619  dom cdm 5621  ran crn 5622   ∘ ccom 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633

This theorem is referenced by:  trclublem  14912  trclubgNEW  43725  trclexi  43727  cnvtrcl0  43733  xpintrreld  43773  trrelsuperreldg  43775  trrelsuperrel2dg  43778