Theorem eliunxp2 42112

Description: Membership in a union of Cartesian products over its second component, analogous to eliunxp 5259. (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 5227	. . . . . . . 8 ⊢ Rel (𝐴 × {𝑦})
2	1	rgenw 2924	. . . . . . 7 ⊢ ∀𝑦 ∈ 𝐵 Rel (𝐴 × {𝑦})
3		reliun 5239	. . . . . . 7 ⊢ (Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∀𝑦 ∈ 𝐵 Rel (𝐴 × {𝑦}))
4	2, 3	mpbir 221	. . . . . 6 ⊢ Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
5		elrel 5222	. . . . . 6 ⊢ ((Rel ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
6	4, 5	mpan 706	. . . . 5 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7		excom 2042	. . . . 5 ⊢ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
8	6, 7	sylibr 224	. . . 4 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) → ∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩)
9	8	pm4.71ri 665	. . 3 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
10		nfiu1 4550	. . . . 5 ⊢ Ⅎ𝑦∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
11	10	nfel2 2781	. . . 4 ⊢ Ⅎ𝑦 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
12	11	19.41 2103	. . 3 ⊢ (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ (∃𝑦∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
13		19.41v 1914	. . . . 5 ⊢ (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ (∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
14		eleq1 2689	. . . . . . . 8 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})))
15		opeliun2xp 42111	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
16		ancom 466	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
17	15, 16	bitri 264	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
18	14, 17	syl6bb 276	. . . . . . 7 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
19	18	pm5.32i 669	. . . . . 6 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
20	19	exbii 1774	. . . . 5 ⊢ (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
21	13, 20	bitr3i 266	. . . 4 ⊢ ((∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
22	21	exbii 1774	. . 3 ⊢ (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})) ↔ ∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
23	9, 12, 22	3bitr2i 288	. 2 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
24		excom 2042	. 2 ⊢ (∃𝑦∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
25	23, 24	bitri 264	1 ⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  {csn 4177  ⟨cop 4183  ∪ ciun 4520   × cxp 5112  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  mpt2mptx2  42113