Theorem txprel 36323

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 36322	. . 3 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V))
2		xpss 5677	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3945	. 2 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × V)
4		df-rel 5668	. 2 ⊢ (Rel (𝐴 ⊗ 𝐵) ↔ (𝐴 ⊗ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 234	1 ⊢ Rel (𝐴 ⊗ 𝐵)

Syntax hints:  Vcvv 3453   ⊆ wss 3904   × cxp 5659  Rel wrel 5666   ⊗ ctxp 36274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-res 5673  df-txp 36298

This theorem is referenced by:  pprodss4v  36328