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Theorem 2ndrn 6035
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 simpr 109 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴𝑅)
2 1st2nd 6033 . . 3 ((Rel 𝑅𝐴𝑅) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
32, 1eqeltrrd 2192 . 2 ((Rel 𝑅𝐴𝑅) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝑅)
4 1stexg 6019 . . . 4 (𝐴𝑅 → (1st𝐴) ∈ V)
5 2ndexg 6020 . . . 4 (𝐴𝑅 → (2nd𝐴) ∈ V)
64, 5jca 302 . . 3 (𝐴𝑅 → ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V))
7 opelrng 4731 . . . 4 (((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V ∧ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝑅) → (2nd𝐴) ∈ ran 𝑅)
873expa 1164 . . 3 ((((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V) ∧ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝑅) → (2nd𝐴) ∈ ran 𝑅)
96, 8sylan 279 . 2 ((𝐴𝑅 ∧ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝑅) → (2nd𝐴) ∈ ran 𝑅)
101, 3, 9syl2anc 406 1 ((Rel 𝑅𝐴𝑅) → (2nd𝐴) ∈ ran 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1463  Vcvv 2657  cop 3496  ran crn 4500  Rel wrel 4504  cfv 5081  1st c1st 5990  2nd c2nd 5991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fo 5087  df-fv 5089  df-1st 5992  df-2nd 5993
This theorem is referenced by: (None)
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