Theorem xptrrel 14907

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 4190	. . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵)
2		dmxpss 6130	. . . . . . . 8 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
3	1, 2	sstri 3944	. . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴
4		inss2 4191	. . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
5		rnxpss 6131	. . . . . . . 8 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
6	4, 5	sstri 3944	. . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵
7	3, 6	ssini 4193	. . . . . 6 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴 ∩ 𝐵)
8		eqimss 3993	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ⊆ ∅)
9	7, 8	sstrid 3946	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅)
10		ss0 4355	. . . . 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 14906	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅)
13		0ss 4353	. . 3 ⊢ ∅ ⊆ (𝐴 × 𝐵)
14	12, 13	eqsstrdi 3979	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
15		neqne 2941	. . . 4 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ≠ ∅)
16	15	xpcoidgend 14902	. . 3 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))
17		ssid 3957	. . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵)
18	16, 17	eqsstrdi 3979	. 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 1542   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286   × cxp 5623  dom cdm 5625  ran crn 5626   ∘ ccom 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637

This theorem is referenced by:  trclublem  14922  trclubgNEW  43895  trclexi  43897  cnvtrcl0  43903  xpintrreld  43943  trrelsuperreldg  43945  trrelsuperrel2dg  43948