Theorem eliunxp2 45557

Description: Membership in a union of Cartesian products over its second component, analogous to eliunxp 5735. (Contributed by AV, 30-Mar-2019.)

Assertion

Ref	Expression
eliunxp2	⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))

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

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

Proof of Theorem eliunxp2

Step	Hyp	Ref	Expression
1		relxp 5598	. . . . . . . 8 ⊢ Rel (𝐴 × {𝑦})
2	1	rgenw 3075	. . . . . . 7 ⊢ ∀𝑦 ∈ 𝐵 Rel (𝐴 × {𝑦})
3		reliun 5715	. . . . . . 7 ⊢ (Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∀𝑦 ∈ 𝐵 Rel (𝐴 × {𝑦}))
4	2, 3	mpbir 230	. . . . . 6 ⊢ Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
5		elrel 5697	. . . . . 6 ⊢ ((Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
6	4, 5	mpan 686	. . . . 5 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7		excom 2164	. . . . 5 ⊢ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
8	6, 7	sylibr 233	. . . 4 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) → ∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩)
9	8	pm4.71ri 560	. . 3 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
10		nfiu1 4955	. . . . 5 ⊢ Ⅎ𝑦∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
11	10	nfel2 2924	. . . 4 ⊢ Ⅎ𝑦 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
12	11	19.41 2231	. . 3 ⊢ (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
13		19.41v 1954	. . . . 5 ⊢ (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ (∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
14		eleq1 2826	. . . . . . . 8 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
15		opeliun2xp 45556	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
16	15	biancomi 462	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
17	14, 16	bitrdi 286	. . . . . . 7 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
18	17	pm5.32i 574	. . . . . 6 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
19	18	exbii 1851	. . . . 5 ⊢ (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
20	13, 19	bitr3i 276	. . . 4 ⊢ ((∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
21	20	exbii 1851	. . 3 ⊢ (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ ∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
22	9, 12, 21	3bitr2i 298	. 2 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
23		excom 2164	. 2 ⊢ (∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
24	22, 23	bitri 274	1 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  {csn 4558  ⟨cop 4564  ∪ ciun 4921   × cxp 5578  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-iun 4923  df-opab 5133  df-xp 5586  df-rel 5587

This theorem is referenced by:  mpomptx2  45558