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Theorem xp1st 5734
 Description: Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
xp1st (A (B × 𝐶) → (1stA) B)

Proof of Theorem xp1st
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4305 . 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, 3op1std 5717 . . . . . 6 (A = ⟨𝑏, 𝑐⟩ → (1stA) = 𝑏)
54eleq1d 2103 . . . . 5 (A = ⟨𝑏, 𝑐⟩ → ((1stA) B𝑏 B))
65biimpar 281 . . . 4 ((A = ⟨𝑏, 𝑐 𝑏 B) → (1stA) B)
76adantrr 448 . . 3 ((A = ⟨𝑏, 𝑐 (𝑏 B 𝑐 𝐶)) → (1stA) B)
87exlimivv 1773 . 2 (𝑏𝑐(A = ⟨𝑏, 𝑐 (𝑏 B 𝑐 𝐶)) → (1stA) B)
91, 8sylbi 114 1 (A (B × 𝐶) → (1stA) B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ⟨cop 3370   × cxp 4286  ‘cfv 4845  1st c1st 5707 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-fv 4853  df-1st 5709 This theorem is referenced by:  dfplpq2  6338  dfmpq2  6339  enqbreq2  6341  enqdc1  6346  mulpipq2  6355  preqlu  6455  elnp1st2nd  6459  cauappcvgprlemladd  6630  elreal2  6728  cnref1o  8357  frecuzrdgrrn  8875
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