Theorem eliunxp 4418

Description: Membership in a union of cross products. Analogue of elxp 4305 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 4390	. . . . . 6 ⊢ Rel ({x} × B)
2	1	rgenw 2370	. . . . 5 ⊢ ∀x ∈ A Rel ({x} × B)
3		reliun 4401	. . . . 5 ⊢ (Rel ∪ x ∈ A ({x} × B) ↔ ∀x ∈ A Rel ({x} × B))
4	2, 3	mpbir 134	. . . 4 ⊢ Rel ∪ x ∈ A ({x} × B)
5		elrel 4385	. . . 4 ⊢ ((Rel ∪ x ∈ A ({x} × B) ∧ 𝐶 ∈ ∪ x ∈ A ({x} × B)) → ∃x∃y 𝐶 = ⟨x, y⟩)
6	4, 5	mpan 400	. . 3 ⊢ (𝐶 ∈ ∪ x ∈ A ({x} × B) → ∃x∃y 𝐶 = ⟨x, y⟩)
7	6	pm4.71ri 372	. 2 ⊢ (𝐶 ∈ ∪ x ∈ A ({x} × B) ↔ (∃x∃y 𝐶 = ⟨x, y⟩ ∧ 𝐶 ∈ ∪ x ∈ A ({x} × B)))
8		nfiu1 3678	. . . 4 ⊢ Ⅎx∪ x ∈ A ({x} × B)
9	8	nfel2 2187	. . 3 ⊢ Ⅎx 𝐶 ∈ ∪ x ∈ A ({x} × B)
10	9	19.41 1573	. 2 ⊢ (∃x(∃y 𝐶 = ⟨x, y⟩ ∧ 𝐶 ∈ ∪ x ∈ A ({x} × B)) ↔ (∃x∃y 𝐶 = ⟨x, y⟩ ∧ 𝐶 ∈ ∪ x ∈ A ({x} × B)))
11		19.41v 1779	. . . 4 ⊢ (∃y(𝐶 = ⟨x, y⟩ ∧ 𝐶 ∈ ∪ x ∈ A ({x} × B)) ↔ (∃y 𝐶 = ⟨x, y⟩ ∧ 𝐶 ∈ ∪ x ∈ A ({x} × B)))
12		eleq1 2097	. . . . . . 7 ⊢ (𝐶 = ⟨x, y⟩ → (𝐶 ∈ ∪ x ∈ A ({x} × B) ↔ ⟨x, y⟩ ∈ ∪ x ∈ A ({x} × B)))
13		opeliunxp 4338	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ ∪ x ∈ A ({x} × B) ↔ (x ∈ A ∧ y ∈ B))
14	12, 13	syl6bb 185	. . . . . 6 ⊢ (𝐶 = ⟨x, y⟩ → (𝐶 ∈ ∪ x ∈ A ({x} × B) ↔ (x ∈ A ∧ y ∈ B)))
15	14	pm5.32i 427	. . . . 5 ⊢ ((𝐶 = ⟨x, y⟩ ∧ 𝐶 ∈ ∪ x ∈ A ({x} × B)) ↔ (𝐶 = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)))
16	15	exbii 1493	. . . 4 ⊢ (∃y(𝐶 = ⟨x, y⟩ ∧ 𝐶 ∈ ∪ x ∈ A ({x} × B)) ↔ ∃y(𝐶 = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)))
17	11, 16	bitr3i 175	. . 3 ⊢ ((∃y 𝐶 = ⟨x, y⟩ ∧ 𝐶 ∈ ∪ x ∈ A ({x} × B)) ↔ ∃y(𝐶 = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)))
18	17	exbii 1493	. 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 197	1 ⊢ (𝐶 ∈ ∪ x ∈ A ({x} × B) ↔ ∃x∃y(𝐶 = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  {csn 3367  ⟨cop 3370  ∪ ciun 3648   × cxp 4286  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-opab 3810  df-xp 4294  df-rel 4295

This theorem is referenced by:  raliunxp  4420  rexiunxp  4421  dfmpt3  4964  mpt2mptx  5537