Theorem eliunxp 5804

Description: Membership in a union of Cartesian products. Analogue of elxp 5664 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 5659	. . . . . 6 ⊢ Rel ({𝑥} × 𝐵)
2	1	rgenw 3049	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)
3		reliun 5782	. . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵))
4	2, 3	mpbir 231	. . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
5		elrel 5764	. . . 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 4994	. . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
9	8	nfel2 2911	. . 3 ⊢ Ⅎ𝑥 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
10	9	19.41 2236	. 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 2817	. . . . . . 7 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
13		opeliunxp 5708	. . . . . . 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 2109  ∀wral 3045  {csn 4592  ⟨cop 4598  ∪ ciun 4958   × cxp 5639  Rel wrel 5646

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-iun 4960  df-opab 5173  df-xp 5647  df-rel 5648

This theorem is referenced by:  raliunxp  5806  dfmpt3  6655  mpomptx  7505  fsumcom2  15747  fprodcom2  15957  isfunc  17833  gsum2d2  19911  dprd2d2  19983  fsumvma  27131  2ndresdju  32580  mpomptxf  32608  gsumwrd2dccatlem  33013  poimirlem26  37647  dvnprodlem1  45951