Theorem opeliunxp 5708

Description: Membership in a union of Cartesian 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		df-iun 4960	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)}
2	1	eleq2i 2821	. 2 ⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)})
3		opex 5427	. . 3 ⊢ ⟨𝑥, 𝐶⟩ ∈ V
4		df-rex 3055	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)))
5		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵))
6		nfs1v 2157	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑥 ∈ 𝐴
7		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑧}
8		nfcsb1v 3889	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
9	7, 8	nfxp 5674	. . . . . . . 8 ⊢ Ⅎ𝑥({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
10	9	nfcri 2884	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
11	6, 10	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
12		sbequ12 2252	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑥 ∈ 𝐴))
13		sneq 4602	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
14		csbeq1a 3879	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
15	13, 14	xpeq12d 5672	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
16	15	eleq2d 2815	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
17	12, 16	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
18	5, 11, 17	cbvexv1 2340	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
19	4, 18	bitri 275	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
20		eleq1 2817	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 𝐶⟩ → (𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
21	20	anbi2d 630	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 𝐶⟩ → (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
22	21	exbidv 1921	. . . 4 ⊢ (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
23	19, 22	bitrid 283	. . 3 ⊢ (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
24	3, 23	elab 3649	. 2 ⊢ (⟨𝑥, 𝐶⟩ ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)} ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
25		opelxp 5677	. . . . . 6 ⊢ (⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑥 ∈ {𝑧} ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵))
26	25	anbi2i 623	. . . . 5 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
27		an12 645	. . . . 5 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
28		velsn 4608	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
29		equcom 2018	. . . . . . 7 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
30	28, 29	bitri 275	. . . . . 6 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
31	30	anbi1i 624	. . . . 5 ⊢ ((𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
32	26, 27, 31	3bitri 297	. . . 4 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
33	32	exbii 1848	. . 3 ⊢ (∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
34		sbequ12r 2253	. . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
35	14	equcoms 2020	. . . . . . 7 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
36	35	eqcomd 2736	. . . . . 6 ⊢ (𝑧 = 𝑥 → ⦋𝑧 / 𝑥⦌𝐵 = 𝐵)
37	36	eleq2d 2815	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐶 ∈ 𝐵))
38	34, 37	anbi12d 632	. . . 4 ⊢ (𝑧 = 𝑥 → (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))
39	38	equsexvw 2005	. . 3 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
40	33, 39	bitri 275	. 2 ⊢ (∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
41	2, 24, 40	3bitri 297	1 ⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779  [wsb 2065   ∈ wcel 2109  {cab 2708  ∃wrex 3054  ⦋csb 3865  {csn 4592  ⟨cop 4598  ∪ ciun 4958   × cxp 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-iun 4960  df-opab 5173  df-xp 5647

This theorem is referenced by:  eliunxp  5804  opeliunxp2  5805  opeliunxp2f  8192  gsum2d2lem  19910  gsum2d2  19911  gsumcom2  19912  dprdval  19942  ptbasfi  23475  cnextfun  23958  cnextfvval  23959  cnextf  23960  dvbsss  25810  iunsnima  32553