Theorem txprel 36047

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 36046	. . 3 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V))
2		xpss 5636	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3926	. 2 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × V)
4		df-rel 5627	. 2 ⊢ (Rel (𝐴 ⊗ 𝐵) ↔ (𝐴 ⊗ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 231	1 ⊢ Rel (𝐴 ⊗ 𝐵)

Syntax hints:  Vcvv 3427   ⊆ wss 3885   × cxp 5618  Rel wrel 5625   ⊗ ctxp 35998

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-ext 2707  ax-sep 5220  ax-pr 5364

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 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-res 5632  df-txp 36022

This theorem is referenced by:  pprodss4v  36052