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Theorem xp2nd 6104
 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 4596 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)))
2 vex 2712 . . . . . . 7 𝑏 ∈ V
3 vex 2712 . . . . . . 7 𝑐 ∈ V
42, 3op2ndd 6087 . . . . . 6 (𝐴 = ⟨𝑏, 𝑐⟩ → (2nd𝐴) = 𝑐)
54eleq1d 2223 . . . . 5 (𝐴 = ⟨𝑏, 𝑐⟩ → ((2nd𝐴) ∈ 𝐶𝑐𝐶))
65biimpar 295 . . . 4 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ 𝑐𝐶) → (2nd𝐴) ∈ 𝐶)
76adantrl 470 . . 3 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)) → (2nd𝐴) ∈ 𝐶)
87exlimivv 1873 . 2 (∃𝑏𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)) → (2nd𝐴) ∈ 𝐶)
91, 8sylbi 120 1 (𝐴 ∈ (𝐵 × 𝐶) → (2nd𝐴) ∈ 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 2125  ⟨cop 3559   × cxp 4577  ‘cfv 5163  2nd c2nd 6077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fv 5171  df-2nd 6079 This theorem is referenced by:  xpf1o  6778  xpmapenlem  6783  djuf1olem  6983  cc2lem  7165  dfplpq2  7253  dfmpq2  7254  enqbreq2  7256  enqdc1  7261  mulpipq2  7270  preqlu  7371  elnp1st2nd  7375  cauappcvgprlemladd  7557  elreal2  7729  cnref1o  9537  frecuzrdgrrn  10285  frec2uzrdg  10286  frecuzrdgrcl  10287  frecuzrdgtcl  10289  frecuzrdgsuc  10291  frecuzrdgrclt  10292  frecuzrdgg  10293  frecuzrdgdomlem  10294  frecuzrdgfunlem  10296  frecuzrdgsuctlem  10300  seq3val  10335  seqvalcd  10336  fisumcom2  11312  fprodcom2fi  11500  eucalgval  11903  eucalginv  11905  eucalglt  11906  eucalgcvga  11907  eucalg  11908  sqpweven  12021  2sqpwodd  12022  ctiunctlemudc  12125  tx1cn  12616  txdis  12624  txhmeo  12666  xmetxp  12854  xmetxpbl  12855  xmettxlem  12856  xmettx  12857
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