Theorem opeliunxp 4322

Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)

Assertion

Ref	Expression
opeliunxp	⊢ (⟨x, 𝐶⟩ ∈ ∪ x ∈ A ({x} × B) ↔ (x ∈ A ∧ 𝐶 ∈ B))

Proof of Theorem opeliunxp

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2543	. 2 ⊢ (⟨x, 𝐶⟩ ∈ ∪ x ∈ A ({x} × B) → ⟨x, 𝐶⟩ ∈ V)
2		vex 2538	. . 3 ⊢ x ∈ V
3		elex 2543	. . . 4 ⊢ (𝐶 ∈ B → 𝐶 ∈ V)
4	3	adantl 262	. . 3 ⊢ ((x ∈ A ∧ 𝐶 ∈ B) → 𝐶 ∈ V)
5		opexgOLD 3939	. . 3 ⊢ ((x ∈ V ∧ 𝐶 ∈ V) → ⟨x, 𝐶⟩ ∈ V)
6	2, 4, 5	sylancr 395	. 2 ⊢ ((x ∈ A ∧ 𝐶 ∈ B) → ⟨x, 𝐶⟩ ∈ V)
7		df-rex 2290	. . . . . 6 ⊢ (∃x ∈ A y ∈ ({x} × B) ↔ ∃x(x ∈ A ∧ y ∈ ({x} × B)))
8		nfv 1402	. . . . . . 7 ⊢ Ⅎz(x ∈ A ∧ y ∈ ({x} × B))
9		nfs1v 1797	. . . . . . . 8 ⊢ Ⅎx[z / x]x ∈ A
10		nfcv 2160	. . . . . . . . . 10 ⊢ Ⅎx{z}
11		nfcsb1v 2859	. . . . . . . . . 10 ⊢ Ⅎx⦋z / x⦌B
12	10, 11	nfxp 4298	. . . . . . . . 9 ⊢ Ⅎx({z} × ⦋z / x⦌B)
13	12	nfcri 2154	. . . . . . . 8 ⊢ Ⅎx y ∈ ({z} × ⦋z / x⦌B)
14	9, 13	nfan 1439	. . . . . . 7 ⊢ Ⅎx([z / x]x ∈ A ∧ y ∈ ({z} × ⦋z / x⦌B))
15		sbequ12 1636	. . . . . . . 8 ⊢ (x = z → (x ∈ A ↔ [z / x]x ∈ A))
16		sneq 3361	. . . . . . . . . 10 ⊢ (x = z → {x} = {z})
17		csbeq1a 2837	. . . . . . . . . 10 ⊢ (x = z → B = ⦋z / x⦌B)
18	16, 17	xpeq12d 4297	. . . . . . . . 9 ⊢ (x = z → ({x} × B) = ({z} × ⦋z / x⦌B))
19	18	eleq2d 2089	. . . . . . . 8 ⊢ (x = z → (y ∈ ({x} × B) ↔ y ∈ ({z} × ⦋z / x⦌B)))
20	15, 19	anbi12d 445	. . . . . . 7 ⊢ (x = z → ((x ∈ A ∧ y ∈ ({x} × B)) ↔ ([z / x]x ∈ A ∧ y ∈ ({z} × ⦋z / x⦌B))))
21	8, 14, 20	cbvex 1621	. . . . . 6 ⊢ (∃x(x ∈ A ∧ y ∈ ({x} × B)) ↔ ∃z([z / x]x ∈ A ∧ y ∈ ({z} × ⦋z / x⦌B)))
22	7, 21	bitri 173	. . . . 5 ⊢ (∃x ∈ A y ∈ ({x} × B) ↔ ∃z([z / x]x ∈ A ∧ y ∈ ({z} × ⦋z / x⦌B)))
23		eleq1 2082	. . . . . . 7 ⊢ (y = ⟨x, 𝐶⟩ → (y ∈ ({z} × ⦋z / x⦌B) ↔ ⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B)))
24	23	anbi2d 440	. . . . . 6 ⊢ (y = ⟨x, 𝐶⟩ → (([z / x]x ∈ A ∧ y ∈ ({z} × ⦋z / x⦌B)) ↔ ([z / x]x ∈ A ∧ ⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B))))
25	24	exbidv 1688	. . . . 5 ⊢ (y = ⟨x, 𝐶⟩ → (∃z([z / x]x ∈ A ∧ y ∈ ({z} × ⦋z / x⦌B)) ↔ ∃z([z / x]x ∈ A ∧ ⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B))))
26	22, 25	syl5bb 181	. . . 4 ⊢ (y = ⟨x, 𝐶⟩ → (∃x ∈ A y ∈ ({x} × B) ↔ ∃z([z / x]x ∈ A ∧ ⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B))))
27		df-iun 3633	. . . 4 ⊢ ∪ x ∈ A ({x} × B) = {y ∣ ∃x ∈ A y ∈ ({x} × B)}
28	26, 27	elab2g 2666	. . 3 ⊢ (⟨x, 𝐶⟩ ∈ V → (⟨x, 𝐶⟩ ∈ ∪ x ∈ A ({x} × B) ↔ ∃z([z / x]x ∈ A ∧ ⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B))))
29		opelxp 4301	. . . . . . 7 ⊢ (⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B) ↔ (x ∈ {z} ∧ 𝐶 ∈ ⦋z / x⦌B))
30	29	anbi2i 433	. . . . . 6 ⊢ (([z / x]x ∈ A ∧ ⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B)) ↔ ([z / x]x ∈ A ∧ (x ∈ {z} ∧ 𝐶 ∈ ⦋z / x⦌B)))
31		an12 483	. . . . . 6 ⊢ (([z / x]x ∈ A ∧ (x ∈ {z} ∧ 𝐶 ∈ ⦋z / x⦌B)) ↔ (x ∈ {z} ∧ ([z / x]x ∈ A ∧ 𝐶 ∈ ⦋z / x⦌B)))
32		elsn 3365	. . . . . . . 8 ⊢ (x ∈ {z} ↔ x = z)
33		equcom 1575	. . . . . . . 8 ⊢ (x = z ↔ z = x)
34	32, 33	bitri 173	. . . . . . 7 ⊢ (x ∈ {z} ↔ z = x)
35	34	anbi1i 434	. . . . . 6 ⊢ ((x ∈ {z} ∧ ([z / x]x ∈ A ∧ 𝐶 ∈ ⦋z / x⦌B)) ↔ (z = x ∧ ([z / x]x ∈ A ∧ 𝐶 ∈ ⦋z / x⦌B)))
36	30, 31, 35	3bitri 195	. . . . 5 ⊢ (([z / x]x ∈ A ∧ ⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B)) ↔ (z = x ∧ ([z / x]x ∈ A ∧ 𝐶 ∈ ⦋z / x⦌B)))
37	36	exbii 1478	. . . 4 ⊢ (∃z([z / x]x ∈ A ∧ ⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B)) ↔ ∃z(z = x ∧ ([z / x]x ∈ A ∧ 𝐶 ∈ ⦋z / x⦌B)))
38		sbequ12r 1637	. . . . . 6 ⊢ (z = x → ([z / x]x ∈ A ↔ x ∈ A))
39	17	equcoms 1576	. . . . . . . 8 ⊢ (z = x → B = ⦋z / x⦌B)
40	39	eqcomd 2027	. . . . . . 7 ⊢ (z = x → ⦋z / x⦌B = B)
41	40	eleq2d 2089	. . . . . 6 ⊢ (z = x → (𝐶 ∈ ⦋z / x⦌B ↔ 𝐶 ∈ B))
42	38, 41	anbi12d 445	. . . . 5 ⊢ (z = x → (([z / x]x ∈ A ∧ 𝐶 ∈ ⦋z / x⦌B) ↔ (x ∈ A ∧ 𝐶 ∈ B)))
43	2, 42	ceqsexv 2570	. . . 4 ⊢ (∃z(z = x ∧ ([z / x]x ∈ A ∧ 𝐶 ∈ ⦋z / x⦌B)) ↔ (x ∈ A ∧ 𝐶 ∈ B))
44	37, 43	bitri 173	. . 3 ⊢ (∃z([z / x]x ∈ A ∧ ⟨x, 𝐶⟩ ∈ ({z} × ⦋z / x⦌B)) ↔ (x ∈ A ∧ 𝐶 ∈ B))
45	28, 44	syl6bb 185	. 2 ⊢ (⟨x, 𝐶⟩ ∈ V → (⟨x, 𝐶⟩ ∈ ∪ x ∈ A ({x} × B) ↔ (x ∈ A ∧ 𝐶 ∈ B)))
46	1, 6, 45	pm5.21nii 607	1 ⊢ (⟨x, 𝐶⟩ ∈ ∪ x ∈ A ({x} × B) ↔ (x ∈ A ∧ 𝐶 ∈ B))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  [wsb 1627  ∃wrex 2285  Vcvv 2535  ⦋csb 2829  {csn 3350  ⟨cop 3353  ∪ ciun 3631   × cxp 4270

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-opab 3793  df-xp 4278

This theorem is referenced by:  eliunxp  4402  opeliunxp2  4403