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Theorem elsnxp 6135
Description: Membership in a Cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.)
Assertion
Ref Expression
elsnxp (𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝑦,𝑋   𝑦,𝑍

Proof of Theorem elsnxp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elxp 5571 . . 3 (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
2 df-rex 3141 . . . . . 6 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
3 an13 643 . . . . . . 7 ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ (𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
43exbii 1839 . . . . . 6 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
52, 4bitr4i 279 . . . . 5 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
6 elsni 4574 . . . . . . . . 9 (𝑥 ∈ {𝑋} → 𝑥 = 𝑋)
76opeq1d 4801 . . . . . . . 8 (𝑥 ∈ {𝑋} → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
87eqeq2d 2829 . . . . . . 7 (𝑥 ∈ {𝑋} → (𝑍 = ⟨𝑥, 𝑦⟩ ↔ 𝑍 = ⟨𝑋, 𝑦⟩))
98biimpa 477 . . . . . 6 ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
109reximi 3240 . . . . 5 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
115, 10sylbir 236 . . . 4 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1211exlimiv 1922 . . 3 (∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
131, 12sylbi 218 . 2 (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
14 snidg 4589 . . . . 5 (𝑋𝑉𝑋 ∈ {𝑋})
15 opelxpi 5585 . . . . 5 ((𝑋 ∈ {𝑋} ∧ 𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
1614, 15sylan 580 . . . 4 ((𝑋𝑉𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
17 eleq1 2897 . . . 4 (𝑍 = ⟨𝑋, 𝑦⟩ → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴)))
1816, 17syl5ibrcom 248 . . 3 ((𝑋𝑉𝑦𝐴) → (𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴)))
1918rexlimdva 3281 . 2 (𝑋𝑉 → (∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴)))
2013, 19impbid2 227 1 (𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  {csn 4557  cop 4563   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-xp 5554
This theorem is referenced by:  esum2dlem  31250  esum2d  31251  projf1o  41335  sge0xp  42588
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