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Theorem xp2nd 5732
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 4304 . 2 (A (B × 𝐶) ↔ 𝑏𝑐(A = ⟨𝑏, 𝑐 (𝑏 B 𝑐 𝐶)))
2 vex 2554 . . . . . . 7 𝑏 V
3 vex 2554 . . . . . . 7 𝑐 V
42, 3op2ndd 5715 . . . . . 6 (A = ⟨𝑏, 𝑐⟩ → (2ndA) = 𝑐)
54eleq1d 2103 . . . . 5 (A = ⟨𝑏, 𝑐⟩ → ((2ndA) 𝐶𝑐 𝐶))
65biimpar 281 . . . 4 ((A = ⟨𝑏, 𝑐 𝑐 𝐶) → (2ndA) 𝐶)
76adantrl 447 . . 3 ((A = ⟨𝑏, 𝑐 (𝑏 B 𝑐 𝐶)) → (2ndA) 𝐶)
87exlimivv 1773 . 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 1242  wex 1378   wcel 1390  cop 3369   × cxp 4285  cfv 4844  2nd c2nd 5705
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 3865  ax-pow 3917  ax-pr 3934  ax-un 4135
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-iota 4809  df-fun 4846  df-fv 4852  df-2nd 5707
This theorem is referenced by:  dfplpq2  6331  dfmpq2  6332  enqbreq2  6334  enqdc1  6339  mulpipq2  6348  preqlu  6447  elnp1st2nd  6451  elreal2  6680
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