Theorem txpss3v 36376

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 36352	. 2 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2		inss1 4188	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴)
3		relco 6109	. . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴)
4		vex 3458	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		vex 3458	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5867	. . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧)
7	4	brresi 5986	. . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧))
8	7	simplbi 501	. . . . . . . 8 ⊢ (𝑦(1st ↾ (V × V))𝑧 → 𝑦 ∈ (V × V))
9	6, 8	sylbi 220	. . . . . . 7 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 → 𝑦 ∈ (V × V))
10	9	adantl 486	. . . . . 6 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
11	10	exlimiv 1959	. . . . 5 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
13	12, 5	opelco 5856	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦))
14		opelxp 5696	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V)))
15	12, 14	mpbiran 721	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ 𝑦 ∈ (V × V))
16	11, 13, 15	3imtr4i 295	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
17	3, 16	relssi 5772	. . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
18	2, 17	sstri 3945	. 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
19	1, 18	eqsstri 3982	1 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V))

This proof depends on syntax axioms:   ∧ wa 400  ∃wex 1808   ∈ wcel 2142  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ⟨cop 4594   class class class wbr 5108   × cxp 5658  ^◡ccnv 5659   ↾ cres 5662   ∘ ccom 5664  1^st c1st 7982  2^nd c2nd 7983   ⊗ ctxp 36328

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  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 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-res 5672  df-txp 36352

This theorem is used by:  txprel  36377  brtxp2  36379  pprodss4v  36382