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Theorem xp1st 5819
 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 4389 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)))
2 vex 2577 . . . . . . 7 𝑏 ∈ V
3 vex 2577 . . . . . . 7 𝑐 ∈ V
42, 3op1std 5802 . . . . . 6 (𝐴 = ⟨𝑏, 𝑐⟩ → (1st𝐴) = 𝑏)
54eleq1d 2122 . . . . 5 (𝐴 = ⟨𝑏, 𝑐⟩ → ((1st𝐴) ∈ 𝐵𝑏𝐵))
65biimpar 285 . . . 4 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ 𝑏𝐵) → (1st𝐴) ∈ 𝐵)
76adantrr 456 . . 3 ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)) → (1st𝐴) ∈ 𝐵)
87exlimivv 1792 . 2 (∃𝑏𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏𝐵𝑐𝐶)) → (1st𝐴) ∈ 𝐵)
91, 8sylbi 118 1 (𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259  ∃wex 1397   ∈ wcel 1409  ⟨cop 3405   × cxp 4370  ‘cfv 4929  1st c1st 5792 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fv 4937  df-1st 5794 This theorem is referenced by:  dfplpq2  6509  dfmpq2  6510  enqbreq2  6512  enqdc1  6517  mulpipq2  6526  preqlu  6627  elnp1st2nd  6631  cauappcvgprlemladd  6813  elreal2  6964  cnref1o  8679  frecuzrdgrrn  9357
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