Theorem txpss3v 36174

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 36150	. 2 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2		inss1 4183	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴)
3		relco 6087	. . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴)
4		vex 3452	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		vex 3452	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5847	. . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧)
7	4	brresi 5967	. . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧))
8	7	simplbi 499	. . . . . . . 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 1944	. . . . 5 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12		vex 3452	. . . . . 6 ⊢ 𝑥 ∈ V
13	12, 5	opelco 5836	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦))
14		opelxp 5676	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V)))
15	12, 14	mpbiran 717	. . . . 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 5752	. . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
18	2, 17	sstri 3940	. 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
19	1, 18	eqsstri 3977	1 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V))

Syntax hints:   ∧ wa 398  ∃wex 1793   ∈ wcel 2136  Vcvv 3448   ∩ cin 3898   ⊆ wss 3899  ⟨cop 4582   class class class wbr 5094   × cxp 5638  ^◡ccnv 5639   ↾ cres 5642   ∘ ccom 5644  1^st c1st 7957  2^nd c2nd 7958   ⊗ ctxp 36126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-res 5652  df-txp 36150

This theorem is referenced by:  txprel  36175  brtxp2  36177  pprodss4v  36180