Theorem eliunxp2 48459

Description: Membership in a union of Cartesian products over its second component, analogous to eliunxp 5781. (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 5637	. . . . . . . 8 ⊢ Rel (𝐴 × {𝑦})
2	1	rgenw 3052	. . . . . . 7 ⊢ ∀𝑦 ∈ 𝐵 Rel (𝐴 × {𝑦})
3		reliun 5760	. . . . . . 7 ⊢ (Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∀𝑦 ∈ 𝐵 Rel (𝐴 × {𝑦}))
4	2, 3	mpbir 231	. . . . . 6 ⊢ Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
5		elrel 5742	. . . . . 6 ⊢ ((Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
6	4, 5	mpan 690	. . . . 5 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7		excom 2167	. . . . 5 ⊢ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
8	6, 7	sylibr 234	. . . 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 4977	. . . . 5 ⊢ Ⅎ𝑦∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
11	10	nfel2 2914	. . . 4 ⊢ Ⅎ𝑦 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
12	11	19.41 2240	. . 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 2821	. . . . . . . 8 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
15		opeliun2xp 5687	. . . . . . . . 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 287	. . . . . . 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 1849	. . . . 5 ⊢ (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
20	13, 19	bitr3i 277	. . . 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 299	. 2 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
23		excom 2167	. 2 ⊢ (∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
24	22, 23	bitri 275	1 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048  {csn 4575  ⟨cop 4581  ∪ ciun 4941   × cxp 5617  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-iun 4943  df-opab 5156  df-xp 5625  df-rel 5626

This theorem is referenced by:  mpomptx2  48460