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Theorem 2ndrn 5732
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 𝑅 A 𝑅) → (2ndA) ran 𝑅)

Proof of Theorem 2ndrn
StepHypRef Expression
1 simpr 103 . 2 ((Rel 𝑅 A 𝑅) → A 𝑅)
2 1st2nd 5730 . . 3 ((Rel 𝑅 A 𝑅) → A = ⟨(1stA), (2ndA)⟩)
32, 1eqeltrrd 2097 . 2 ((Rel 𝑅 A 𝑅) → ⟨(1stA), (2ndA)⟩ 𝑅)
4 1stexg 5717 . . . 4 (A 𝑅 → (1stA) V)
5 2ndexg 5718 . . . 4 (A 𝑅 → (2ndA) V)
64, 5jca 290 . . 3 (A 𝑅 → ((1stA) V (2ndA) V))
7 opelrng 4493 . . . 4 (((1stA) V (2ndA) V ⟨(1stA), (2ndA)⟩ 𝑅) → (2ndA) ran 𝑅)
873expa 1090 . . 3 ((((1stA) V (2ndA) V) ⟨(1stA), (2ndA)⟩ 𝑅) → (2ndA) ran 𝑅)
96, 8sylan 267 . 2 ((A 𝑅 ⟨(1stA), (2ndA)⟩ 𝑅) → (2ndA) ran 𝑅)
101, 3, 9syl2anc 393 1 ((Rel 𝑅 A 𝑅) → (2ndA) ran 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1374  Vcvv 2535  cop 3353  ran crn 4273  Rel wrel 4277  cfv 4829  1st c1st 5688  2nd c2nd 5689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690  df-2nd 5691
This theorem is referenced by: (None)
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