Theorem eliunxp 5821

Description: Membership in a union of Cartesian products. Analogue of elxp 5682 for nonconstant 𝐵(𝑥). (Contributed by Mario Carneiro, 29-Dec-2014.)

Assertion

Ref	Expression
eliunxp	⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem eliunxp

Step	Hyp	Ref	Expression
1		relxp 5677	. . . . . 6 ⊢ Rel ({𝑥} × 𝐵)
2	1	rgenw 3082	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)
3		reliun 5801	. . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵))
4	2, 3	mpbir 234	. . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
5		elrel 5782	. . . 4 ⊢ ((Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
6	4, 5	mpan 703	. . 3 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7	6	pm4.71ri 570	. 2 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
8		nfiu1 4990	. . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
9	8	nfel2 2942	. . 3 ⊢ Ⅎ𝑥 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
10	9	19.41 2273	. 2 ⊢ (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
11		19.41v 1982	. . . 4 ⊢ (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
12		eleq1 2850	. . . . . . 7 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
13		opeliunxp 5726	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
14	12, 13	bitrdi 290	. . . . . 6 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
15	14	pm5.32i 585	. . . . 5 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
16	15	exbii 1881	. . . 4 ⊢ (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
17	11, 16	bitr3i 280	. . 3 ⊢ ((∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
18	17	exbii 1881	. 2 ⊢ (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
19	7, 10, 18	3bitr2i 302	1 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3078  {csn 4587  ⟨cop 4593  ∪ ciun 4954   × cxp 5657  Rel wrel 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-iun 4956  df-opab 5172  df-xp 5665  df-rel 5666

This theorem is used by:  raliunxp  5823  dfmpt3  6670  mpomptx  7530  fsumcom2  15864  fprodcom2  16077  isfunc  17959  gsum2d2  20107  dprd2d2  20179  fsumvma  27457  2ndresdju  33130  mpomptxf  33159  gsumwrd2dccatlem  33525  poimirlem26  38403  dvnprodlem1  46782