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Theorem xp1st 6031
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 4526 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)))
2 vex 2663 . . . . . . 7 𝑏 ∈ V
3 vex 2663 . . . . . . 7 𝑐 ∈ V
42, 3op1std 6014 . . . . . 6 (𝐴 = ⟨𝑏, 𝑐⟩ → (1st𝐴) = 𝑏)
54eleq1d 2186 . . . . 5 (𝐴 = ⟨𝑏, 𝑐⟩ → ((1st𝐴) ∈ 𝐵𝑏𝐵))
65biimpar 295 . . . 4 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ 𝑏𝐵) → (1st𝐴) ∈ 𝐵)
76adantrr 470 . . 3 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)) → (1st𝐴) ∈ 𝐵)
87exlimivv 1852 . 2 (∃𝑏𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)) → (1st𝐴) ∈ 𝐵)
91, 8sylbi 120 1 (𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wex 1453  wcel 1465  cop 3500   × cxp 4507  cfv 5093  1st c1st 6004
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101  df-1st 6006
This theorem is referenced by:  disjxp1  6101  xpf1o  6706  xpmapenlem  6711  djuf1olem  6906  eldju1st  6924  dfplpq2  7130  dfmpq2  7131  enqbreq2  7133  enqdc1  7138  mulpipq2  7147  preqlu  7248  elnp1st2nd  7252  cauappcvgprlemladd  7434  elreal2  7606  cnref1o  9408  frecuzrdgrrn  10149  frec2uzrdg  10150  frecuzrdgrcl  10151  frecuzrdgsuc  10155  frecuzrdgrclt  10156  frecuzrdgg  10157  frecuzrdgsuctlem  10164  seq3val  10199  seqvalcd  10200  fsum2dlemstep  11171  fisumcom2  11175  eucalgval  11662  eucalginv  11664  eucalglt  11665  eucalg  11667  sqpweven  11780  2sqpwodd  11781  ctiunctlemudc  11877  tx2cn  12366  txdis  12373  txhmeo  12415  xmetxp  12603  xmetxpbl  12604  xmettxlem  12605  xmettx  12606
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