Theorem txprel 34882

Description: A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
txprel	⊢ Rel (𝐴 ⊗ 𝐵)

Proof of Theorem txprel

Step	Hyp	Ref	Expression
1		txpss3v 34881	. . 3 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V))
2		xpss 5693	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3992	. 2 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × V)
4		df-rel 5684	. 2 ⊢ (Rel (𝐴 ⊗ 𝐵) ↔ (𝐴 ⊗ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 230	1 ⊢ Rel (𝐴 ⊗ 𝐵)

Syntax hints:  Vcvv 3475   ⊆ wss 3949   × cxp 5675  Rel wrel 5682   ⊗ ctxp 34833

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-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-res 5689  df-txp 34857

This theorem is referenced by:  pprodss4v  34887