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Theorem 2ndrn 8031
Description: The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
2ndrn ((Rel 𝑅𝐴𝑅) → (2nd𝐴) ∈ ran 𝑅)

Proof of Theorem 2ndrn
StepHypRef Expression
1 1st2nd 8029 . . 3 ((Rel 𝑅𝐴𝑅) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 simpr 484 . . 3 ((Rel 𝑅𝐴𝑅) → 𝐴𝑅)
31, 2eqeltrrd 2833 . 2 ((Rel 𝑅𝐴𝑅) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝑅)
4 fvex 6904 . . 3 (1st𝐴) ∈ V
5 fvex 6904 . . 3 (2nd𝐴) ∈ V
64, 5opelrn 5942 . 2 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝑅 → (2nd𝐴) ∈ ran 𝑅)
73, 6syl 17 1 ((Rel 𝑅𝐴𝑅) → (2nd𝐴) ∈ ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  cop 4634  ran crn 5677  Rel wrel 5681  cfv 6543  1st c1st 7977  2nd c2nd 7978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7979  df-2nd 7980
This theorem is referenced by:  gsumhashmul  32645  heicant  36989  mblfinlem1  36991
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