Theorem eliunxp 4821

Description: Membership in a union of cross products. Analogue of elxp 4696 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 4788	. . . . . 6 ⊢ Rel ({𝑥} × 𝐵)
2	1	rgenw 2562	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)
3		reliun 4800	. . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵))
4	2, 3	mpbir 146	. . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
5		elrel 4781	. . . 4 ⊢ ((Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
6	4, 5	mpan 424	. . 3 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7	6	pm4.71ri 392	. 2 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
8		nfiu1 3959	. . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
9	8	nfel2 2362	. . 3 ⊢ Ⅎ𝑥 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
10	9	19.41 1710	. 2 ⊢ (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
11		19.41v 1927	. . . 4 ⊢ (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
12		eleq1 2269	. . . . . . 7 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
13		opeliunxp 4734	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
14	12, 13	bitrdi 196	. . . . . 6 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
15	14	pm5.32i 454	. . . . 5 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
16	15	exbii 1629	. . . 4 ⊢ (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
17	11, 16	bitr3i 186	. . 3 ⊢ ((∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
18	17	exbii 1629	. 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 208	1 ⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  {csn 3634  ⟨cop 3637  ∪ ciun 3929   × cxp 4677  Rel wrel 4684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-iun 3931  df-opab 4110  df-xp 4685  df-rel 4686

This theorem is referenced by:  raliunxp  4823  rexiunxp  4824  dfmpt3  5404  mpomptx  6043  fisumcom2  11793  fprodcom2fi  11981