Theorem eliunxp2 44735

Description: Membership in a union of Cartesian products over its second component, analogous to eliunxp 5672. (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 5537	. . . . . . . 8 ⊢ Rel (𝐴 × {𝑦})
2	1	rgenw 3118	. . . . . . 7 ⊢ ∀𝑦 ∈ 𝐵 Rel (𝐴 × {𝑦})
3		reliun 5653	. . . . . . 7 ⊢ (Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∀𝑦 ∈ 𝐵 Rel (𝐴 × {𝑦}))
4	2, 3	mpbir 234	. . . . . 6 ⊢ Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
5		elrel 5635	. . . . . 6 ⊢ ((Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
6	4, 5	mpan 689	. . . . 5 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7		excom 2166	. . . . 5 ⊢ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
8	6, 7	sylibr 237	. . . 4 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) → ∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩)
9	8	pm4.71ri 564	. . 3 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
10		nfiu1 4915	. . . . 5 ⊢ Ⅎ𝑦∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
11	10	nfel2 2973	. . . 4 ⊢ Ⅎ𝑦 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
12	11	19.41 2235	. . 3 ⊢ (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
13		19.41v 1950	. . . . 5 ⊢ (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ (∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
14		eleq1 2877	. . . . . . . 8 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
15		opeliun2xp 44734	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
16	15	biancomi 466	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
17	14, 16	syl6bb 290	. . . . . . 7 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
18	17	pm5.32i 578	. . . . . 6 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
19	18	exbii 1849	. . . . 5 ⊢ (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
20	13, 19	bitr3i 280	. . . 4 ⊢ ((∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
21	20	exbii 1849	. . 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 302	. 2 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
23		excom 2166	. 2 ⊢ (∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
24	22, 23	bitri 278	1 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  {csn 4525  ⟨cop 4531  ∪ ciun 4881   × cxp 5517  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4883  df-opab 5093  df-xp 5525  df-rel 5526

This theorem is referenced by:  mpomptx2  44736