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Theorem xrnss3v 35781
Description: A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 33449 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33449. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
xrnss3v (𝐴𝐵) ⊆ (V × (V × V))

Proof of Theorem xrnss3v
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xrn 35780 . 2 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
2 inss1 4155 . . 3 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ ((1st ↾ (V × V)) ∘ 𝐴)
3 relco 6064 . . . 4 Rel ((1st ↾ (V × V)) ∘ 𝐴)
4 vex 3444 . . . . . . . . 9 𝑧 ∈ V
5 vex 3444 . . . . . . . . 9 𝑦 ∈ V
64, 5brcnv 5717 . . . . . . . 8 (𝑧(1st ↾ (V × V))𝑦𝑦(1st ↾ (V × V))𝑧)
74brresi 5827 . . . . . . . . 9 (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧))
87simplbi 501 . . . . . . . 8 (𝑦(1st ↾ (V × V))𝑧𝑦 ∈ (V × V))
96, 8sylbi 220 . . . . . . 7 (𝑧(1st ↾ (V × V))𝑦𝑦 ∈ (V × V))
109adantl 485 . . . . . 6 ((𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
1110exlimiv 1931 . . . . 5 (∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12 vex 3444 . . . . . 6 𝑥 ∈ V
1312, 5opelco 5706 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ((1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦))
14 opelxp 5555 . . . . . 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 295 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((1st ↾ (V × V)) ∘ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
173, 16relssi 5624 . . 3 ((1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
182, 17sstri 3924 . 2 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
191, 18eqsstri 3949 1 (𝐴𝐵) ⊆ (V × (V × V))
Colors of variables: wff setvar class
Syntax hints:  wa 399  wex 1781  wcel 2111  Vcvv 3441  cin 3880  wss 3881  cop 4531   class class class wbr 5030   × cxp 5517  ccnv 5518  cres 5521  ccom 5523  1st c1st 7669  2nd c2nd 7670  cxrn 35609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-res 5531  df-xrn 35780
This theorem is referenced by:  xrnrel  35782  brxrn2  35784
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