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Theorem txpss3v 31155
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 31130 . 2 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
2 inss1 3795 . . 3 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ ((1st ↾ (V × V)) ∘ 𝐴)
3 relco 5550 . . . 4 Rel ((1st ↾ (V × V)) ∘ 𝐴)
4 vex 3176 . . . . . . . . 9 𝑧 ∈ V
5 vex 3176 . . . . . . . . 9 𝑦 ∈ V
64, 5brcnv 5227 . . . . . . . 8 (𝑧(1st ↾ (V × V))𝑦𝑦(1st ↾ (V × V))𝑧)
74brres 5323 . . . . . . . . 9 (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦1st 𝑧𝑦 ∈ (V × V)))
87simprbi 479 . . . . . . . 8 (𝑦(1st ↾ (V × V))𝑧𝑦 ∈ (V × V))
96, 8sylbi 206 . . . . . . 7 (𝑧(1st ↾ (V × V))𝑦𝑦 ∈ (V × V))
109adantl 481 . . . . . 6 ((𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
1110exlimiv 1845 . . . . 5 (∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12 vex 3176 . . . . . 6 𝑥 ∈ V
1312, 5opelco 5215 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ((1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦))
14 opelxp 5070 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V)))
1512, 14mpbiran 955 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ 𝑦 ∈ (V × V))
1611, 13, 153imtr4i 280 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((1st ↾ (V × V)) ∘ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
173, 16relssi 5134 . . 3 ((1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
182, 17sstri 3577 . 2 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
191, 18eqsstri 3598 1 (𝐴𝐵) ⊆ (V × (V × V))
Colors of variables: wff setvar class
Syntax hints:  wa 383  wex 1695  wcel 1977  Vcvv 3173  cin 3539  wss 3540  cop 4131   class class class wbr 4583   × cxp 5036  ccnv 5037  cres 5040  ccom 5042  1st c1st 7057  2nd c2nd 7058  ctxp 31106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-res 5050  df-txp 31130
This theorem is referenced by:  txprel  31156  brtxp2  31158  pprodss4v  31161
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