Theorem txpss3v 33339

Description: A tail Cartesian product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
txpss3v	⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V))

Proof of Theorem txpss3v

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-txp 33315	. 2 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2		inss1 4205	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴)
3		relco 6097	. . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴)
4		vex 3497	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		vex 3497	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5753	. . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧)
7	4	brresi 5862	. . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧))
8	7	simplbi 500	. . . . . . . 8 ⊢ (𝑦(1st ↾ (V × V))𝑧 → 𝑦 ∈ (V × V))
9	6, 8	sylbi 219	. . . . . . 7 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 → 𝑦 ∈ (V × V))
10	9	adantl 484	. . . . . 6 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
11	10	exlimiv 1931	. . . . 5 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12		vex 3497	. . . . . 6 ⊢ 𝑥 ∈ V
13	12, 5	opelco 5742	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦))
14		opelxp 5591	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V)))
15	12, 14	mpbiran 707	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ 𝑦 ∈ (V × V))
16	11, 13, 15	3imtr4i 294	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
17	3, 16	relssi 5660	. . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
18	2, 17	sstri 3976	. 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
19	1, 18	eqsstri 4001	1 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V))

Syntax hints:   ∧ wa 398  ∃wex 1780   ∈ wcel 2114  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  ⟨cop 4573   class class class wbr 5066   × cxp 5553  ^◡ccnv 5554   ↾ cres 5557   ∘ ccom 5559  1^st c1st 7687  2^nd c2nd 7688   ⊗ ctxp 33291

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-res 5567  df-txp 33315

This theorem is referenced by:  txprel  33340  brtxp2  33342  pprodss4v  33345