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Theorem xp2nd 5793
 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 4362 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)))
2 vex 2560 . . . . . . 7 𝑏 ∈ V
3 vex 2560 . . . . . . 7 𝑐 ∈ V
42, 3op2ndd 5776 . . . . . 6 (𝐴 = ⟨𝑏, 𝑐⟩ → (2nd𝐴) = 𝑐)
54eleq1d 2106 . . . . 5 (𝐴 = ⟨𝑏, 𝑐⟩ → ((2nd𝐴) ∈ 𝐶𝑐𝐶))
65biimpar 281 . . . 4 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ 𝑐𝐶) → (2nd𝐴) ∈ 𝐶)
76adantrl 447 . . 3 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)) → (2nd𝐴) ∈ 𝐶)
87exlimivv 1776 . 2 (∃𝑏𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)) → (2nd𝐴) ∈ 𝐶)
91, 8sylbi 114 1 (𝐴 ∈ (𝐵 × 𝐶) → (2nd𝐴) ∈ 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ⟨cop 3378   × cxp 4343  ‘cfv 4902  2nd c2nd 5766 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-2nd 5768 This theorem is referenced by:  dfplpq2  6452  dfmpq2  6453  enqbreq2  6455  enqdc1  6460  mulpipq2  6469  preqlu  6570  elnp1st2nd  6574  cauappcvgprlemladd  6756  elreal2  6907  cnref1o  8582  frecuzrdgrrn  9194  frec2uzrdg  9195  frecuzrdgfn  9198  frecuzrdgcl  9199  frecuzrdgsuc  9201
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