Theorem eliunxp 5817

Description: Membership in a union of Cartesian products. Analogue of elxp 5677 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 5672	. . . . . 6 ⊢ Rel ({𝑥} × 𝐵)
2	1	rgenw 3055	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)
3		reliun 5795	. . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵))
4	2, 3	mpbir 231	. . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
5		elrel 5777	. . . 4 ⊢ ((Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
6	4, 5	mpan 690	. . 3 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7	6	pm4.71ri 560	. 2 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
8		nfiu1 5003	. . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
9	8	nfel2 2917	. . 3 ⊢ Ⅎ𝑥 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
10	9	19.41 2235	. 2 ⊢ (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
11		19.41v 1949	. . . 4 ⊢ (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
12		eleq1 2822	. . . . . . 7 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
13		opeliunxp 5721	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
14	12, 13	bitrdi 287	. . . . . 6 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
15	14	pm5.32i 574	. . . . 5 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
16	15	exbii 1848	. . . 4 ⊢ (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
17	11, 16	bitr3i 277	. . 3 ⊢ ((∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
18	17	exbii 1848	. 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 299	1 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3051  {csn 4601  ⟨cop 4607  ∪ ciun 4967   × cxp 5652  Rel wrel 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-iun 4969  df-opab 5182  df-xp 5660  df-rel 5661

This theorem is referenced by:  raliunxp  5819  dfmpt3  6672  mpomptx  7520  fsumcom2  15790  fprodcom2  16000  isfunc  17877  gsum2d2  19955  dprd2d2  20027  fsumvma  27176  2ndresdju  32627  mpomptxf  32655  gsumwrd2dccatlem  33060  poimirlem26  37670  dvnprodlem1  45975