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Theorem xrnss3v 34457
Description: A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 32291 with a different symbol, cf. https://github.com/metamath/set.mm/issues/2469. (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 34456 . 2 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
2 inss1 3976 . . 3 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ ((1st ↾ (V × V)) ∘ 𝐴)
3 relco 5794 . . . 4 Rel ((1st ↾ (V × V)) ∘ 𝐴)
4 vex 3343 . . . . . . . . 9 𝑧 ∈ V
5 vex 3343 . . . . . . . . 9 𝑦 ∈ V
64, 5brcnv 5460 . . . . . . . 8 (𝑧(1st ↾ (V × V))𝑦𝑦(1st ↾ (V × V))𝑧)
74brres 5560 . . . . . . . . 9 (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦1st 𝑧𝑦 ∈ (V × V)))
87simprbi 483 . . . . . . . 8 (𝑦(1st ↾ (V × V))𝑧𝑦 ∈ (V × V))
96, 8sylbi 207 . . . . . . 7 (𝑧(1st ↾ (V × V))𝑦𝑦 ∈ (V × V))
109adantl 473 . . . . . 6 ((𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
1110exlimiv 2007 . . . . 5 (∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12 vex 3343 . . . . . 6 𝑥 ∈ V
1312, 5opelco 5449 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ((1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧𝑧(1st ↾ (V × V))𝑦))
14 opelxp 5303 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V)))
1512, 14mpbiran 991 . . . . 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 5368 . . 3 ((1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
182, 17sstri 3753 . 2 (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
191, 18eqsstri 3776 1 (𝐴𝐵) ⊆ (V × (V × V))
Colors of variables: wff setvar class
Syntax hints:  wa 383  wex 1853  wcel 2139  Vcvv 3340  cin 3714  wss 3715  cop 4327   class class class wbr 4804   × cxp 5264  ccnv 5265  cres 5268  ccom 5270  1st c1st 7331  2nd c2nd 7332  cxrn 34295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-res 5278  df-xrn 34456
This theorem is referenced by:  xrnrel  34458  brxrn2  34460
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