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Theorem xp2nd 5716
Description: Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
xp2nd (A (B × 𝐶) → (2ndA) 𝐶)

Proof of Theorem xp2nd
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4289 . 2 (A (B × 𝐶) ↔ 𝑏𝑐(A = ⟨𝑏, 𝑐 (𝑏 B 𝑐 𝐶)))
2 vex 2538 . . . . . . 7 𝑏 V
3 vex 2538 . . . . . . 7 𝑐 V
42, 3op2ndd 5699 . . . . . 6 (A = ⟨𝑏, 𝑐⟩ → (2ndA) = 𝑐)
54eleq1d 2088 . . . . 5 (A = ⟨𝑏, 𝑐⟩ → ((2ndA) 𝐶𝑐 𝐶))
65biimpar 281 . . . 4 ((A = ⟨𝑏, 𝑐 𝑐 𝐶) → (2ndA) 𝐶)
76adantrl 450 . . 3 ((A = ⟨𝑏, 𝑐 (𝑏 B 𝑐 𝐶)) → (2ndA) 𝐶)
87exlimivv 1758 . 2 (𝑏𝑐(A = ⟨𝑏, 𝑐 (𝑏 B 𝑐 𝐶)) → (2ndA) 𝐶)
91, 8sylbi 114 1 (A (B × 𝐶) → (2ndA) 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wex 1362   wcel 1374  cop 3353   × cxp 4270  cfv 4829  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-fv 4837  df-2nd 5691
This theorem is referenced by:  dfplpq2  6213  dfmpq2  6214  enqbreq2  6216  enqdc1  6221  mulpipq2  6230  preqlu  6326  elnp1st2nd  6330  elreal2  6542
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