Theorem elsnxp 5594

Description: Elementhood to a cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
elsnxp	⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		elxp 5055	. . 3 ⊢ (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)))
2		df-rex 2902	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
3		an13 836	. . . . . . 7 ⊢ ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
4	3	exbii 1764	. . . . . 6 ⊢ (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
5	2, 4	bitr4i 266	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)))
6		elsni 4142	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑋} → 𝑥 = 𝑋)
7	6	opeq1d 4346	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑋} → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
8	7	eqeq2d 2620	. . . . . . 7 ⊢ (𝑥 ∈ {𝑋} → (𝑍 = ⟨𝑥, 𝑦⟩ ↔ 𝑍 = ⟨𝑋, 𝑦⟩))
9	8	biimpa 500	. . . . . 6 ⊢ ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
10	9	reximi 2994	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
11	5, 10	sylbir 224	. . . 4 ⊢ (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
12	11	exlimiv 1845	. . 3 ⊢ (∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
13	1, 12	sylbi 206	. 2 ⊢ (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
14		snidg 4153	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
15		opelxpi 5072	. . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
16	14, 15	sylan 487	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
17		eleq1 2676	. . . 4 ⊢ (𝑍 = ⟨𝑋, 𝑦⟩ → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴)))
18	16, 17	syl5ibrcom 236	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴)))
19	18	rexlimdva 3013	. 2 ⊢ (𝑋 ∈ 𝑉 → (∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴)))
20	13, 19	impbid2 215	1 ⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {csn 4125  ⟨cop 4131   × cxp 5036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044

This theorem is referenced by:  esum2dlem  29481  esum2d  29482  projf1o  38381  sge0xp  39322