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Theorem txpss3v 34850
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 34826 . 2 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
2 inss1 4229 . . 3 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ ((1st ↾ (V × V)) ∘ 𝐴)
3 relco 6108 . . . 4 Rel ((1st ↾ (V × V)) ∘ 𝐴)
4 vex 3479 . . . . . . . . 9 𝑧 ∈ V
5 vex 3479 . . . . . . . . 9 𝑦 ∈ V
64, 5brcnv 5883 . . . . . . . 8 (𝑧(1st ↾ (V × V))𝑦𝑦(1st ↾ (V × V))𝑧)
74brresi 5991 . . . . . . . . 9 (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧))
87simplbi 499 . . . . . . . 8 (𝑦(1st ↾ (V × V))𝑧𝑦 ∈ (V × V))
96, 8sylbi 216 . . . . . . 7 (𝑧(1st ↾ (V × V))𝑦𝑦 ∈ (V × V))
109adantl 483 . . . . . 6 ((𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
1110exlimiv 1934 . . . . 5 (∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12 vex 3479 . . . . . 6 𝑥 ∈ V
1312, 5opelco 5872 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ((1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦))
14 opelxp 5713 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V)))
1512, 14mpbiran 708 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ 𝑦 ∈ (V × V))
1611, 13, 153imtr4i 292 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((1st ↾ (V × V)) ∘ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
173, 16relssi 5788 . . 3 ((1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
182, 17sstri 3992 . 2 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
191, 18eqsstri 4017 1 (𝐴𝐵) ⊆ (V × (V × V))
Colors of variables: wff setvar class
Syntax hints:  wa 397  wex 1782  wcel 2107  Vcvv 3475  cin 3948  wss 3949  cop 4635   class class class wbr 5149   × cxp 5675  ccnv 5676  cres 5679  ccom 5681  1st c1st 7973  2nd c2nd 7974  ctxp 34802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-res 5689  df-txp 34826
This theorem is referenced by:  txprel  34851  brtxp2  34853  pprodss4v  34856
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