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Theorem 2ndrn 5751
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 5749 . . 3 ((Rel 𝑅 A 𝑅) → A = ⟨(1stA), (2ndA)⟩)
32, 1eqeltrrd 2112 . 2 ((Rel 𝑅 A 𝑅) → ⟨(1stA), (2ndA)⟩ 𝑅)
4 1stexg 5736 . . . 4 (A 𝑅 → (1stA) V)
5 2ndexg 5737 . . . 4 (A 𝑅 → (2ndA) V)
64, 5jca 290 . . 3 (A 𝑅 → ((1stA) V (2ndA) V))
7 opelrng 4509 . . . 4 (((1stA) V (2ndA) V ⟨(1stA), (2ndA)⟩ 𝑅) → (2ndA) ran 𝑅)
873expa 1103 . . 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 391 1 ((Rel 𝑅 A 𝑅) → (2ndA) ran 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551  cop 3370  ran crn 4289  Rel wrel 4293  cfv 4845  1st c1st 5707  2nd c2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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