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Theorem xp1st 6070
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 4563 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)))
2 vex 2692 . . . . . . 7 𝑏 ∈ V
3 vex 2692 . . . . . . 7 𝑐 ∈ V
42, 3op1std 6053 . . . . . 6 (𝐴 = ⟨𝑏, 𝑐⟩ → (1st𝐴) = 𝑏)
54eleq1d 2209 . . . . 5 (𝐴 = ⟨𝑏, 𝑐⟩ → ((1st𝐴) ∈ 𝐵𝑏𝐵))
65biimpar 295 . . . 4 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ 𝑏𝐵) → (1st𝐴) ∈ 𝐵)
76adantrr 471 . . 3 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)) → (1st𝐴) ∈ 𝐵)
87exlimivv 1869 . 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 1332  wex 1469  wcel 1481  cop 3534   × cxp 4544  cfv 5130  1st c1st 6043
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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fv 5138  df-1st 6045
This theorem is referenced by:  disjxp1  6140  xpf1o  6745  xpmapenlem  6750  djuf1olem  6945  eldju1st  6963  dfplpq2  7185  dfmpq2  7186  enqbreq2  7188  enqdc1  7193  mulpipq2  7202  preqlu  7303  elnp1st2nd  7307  cauappcvgprlemladd  7489  elreal2  7661  cnref1o  9468  frecuzrdgrrn  10211  frec2uzrdg  10212  frecuzrdgrcl  10213  frecuzrdgsuc  10217  frecuzrdgrclt  10218  frecuzrdgg  10219  frecuzrdgsuctlem  10226  seq3val  10261  seqvalcd  10262  fsum2dlemstep  11234  fisumcom2  11238  eucalgval  11769  eucalginv  11771  eucalglt  11772  eucalg  11774  sqpweven  11887  2sqpwodd  11888  ctiunctlemudc  11984  tx2cn  12476  txdis  12483  txhmeo  12525  xmetxp  12713  xmetxpbl  12714  xmettxlem  12715  xmettx  12716
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