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Theorem txpss3v 32110
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 (𝐴𝐵) ⊆ (V × (V × V))

Proof of Theorem txpss3v
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-txp 32086 . 2 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
2 inss1 3866 . . 3 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ ((1st ↾ (V × V)) ∘ 𝐴)
3 relco 5671 . . . 4 Rel ((1st ↾ (V × V)) ∘ 𝐴)
4 vex 3234 . . . . . . . . 9 𝑧 ∈ V
5 vex 3234 . . . . . . . . 9 𝑦 ∈ V
64, 5brcnv 5337 . . . . . . . 8 (𝑧(1st ↾ (V × V))𝑦𝑦(1st ↾ (V × V))𝑧)
74brres 5437 . . . . . . . . 9 (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦1st 𝑧𝑦 ∈ (V × V)))
87simprbi 479 . . . . . . . 8 (𝑦(1st ↾ (V × V))𝑧𝑦 ∈ (V × V))
96, 8sylbi 207 . . . . . . 7 (𝑧(1st ↾ (V × V))𝑦𝑦 ∈ (V × V))
109adantl 481 . . . . . 6 ((𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
1110exlimiv 1898 . . . . 5 (∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12 vex 3234 . . . . . 6 𝑥 ∈ V
1312, 5opelco 5326 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ((1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦))
14 opelxp 5180 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V)))
1512, 14mpbiran 973 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ 𝑦 ∈ (V × V))
1611, 13, 153imtr4i 281 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((1st ↾ (V × V)) ∘ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
173, 16relssi 5245 . . 3 ((1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
182, 17sstri 3645 . 2 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
191, 18eqsstri 3668 1 (𝐴𝐵) ⊆ (V × (V × V))
Colors of variables: wff setvar class
Syntax hints:  wa 383  wex 1744  wcel 2030  Vcvv 3231  cin 3606  wss 3607  cop 4216   class class class wbr 4685   × cxp 5141  ccnv 5142  cres 5145  ccom 5147  1st c1st 7208  2nd c2nd 7209  ctxp 32062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-res 5155  df-txp 32086
This theorem is referenced by:  txprel  32111  brtxp2  32113  pprodss4v  32116
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