Theorem opeliunxp 4675

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 2746	. 2 ⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ⟨𝑥, 𝐶⟩ ∈ V)
2		opexg 4222	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ⟨𝑥, 𝐶⟩ ∈ V)
3		df-rex 2459	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)))
4		nfv 1526	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵))
5		nfs1v 1937	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑥 ∈ 𝐴
6		nfcv 2317	. . . . . . . . . 10 ⊢ Ⅎ𝑥{𝑧}
7		nfcsb1v 3088	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
8	6, 7	nfxp 4647	. . . . . . . . 9 ⊢ Ⅎ𝑥({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
9	8	nfcri 2311	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
10	5, 9	nfan 1563	. . . . . . 7 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
11		sbequ12 1769	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑥 ∈ 𝐴))
12		sneq 3600	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
13		csbeq1a 3064	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
14	12, 13	xpeq12d 4645	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
15	14	eleq2d 2245	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
16	11, 15	anbi12d 473	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
17	4, 10, 16	cbvex 1754	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
18	3, 17	bitri 184	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
19		eleq1 2238	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝐶⟩ → (𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
20	19	anbi2d 464	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 𝐶⟩ → (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
21	20	exbidv 1823	. . . . 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))))
22	18, 21	bitrid 192	. . . 4 ⊢ (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
23		df-iun 3884	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)}
24	22, 23	elab2g 2882	. . 3 ⊢ (⟨𝑥, 𝐶⟩ ∈ V → (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
25		opelxp 4650	. . . . . . 7 ⊢ (⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑥 ∈ {𝑧} ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵))
26	25	anbi2i 457	. . . . . 6 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
27		an12 561	. . . . . 6 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
28		velsn 3606	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
29		equcom 1704	. . . . . . . 8 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
30	28, 29	bitri 184	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
31	30	anbi1i 458	. . . . . 6 ⊢ ((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 206	. . . . 5 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
33	32	exbii 1603	. . . 4 ⊢ (∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
34		vex 2738	. . . . 5 ⊢ 𝑥 ∈ V
35		sbequ12r 1770	. . . . . 6 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
36	13	equcoms 1706	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
37	36	eqcomd 2181	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ⦋𝑧 / 𝑥⦌𝐵 = 𝐵)
38	37	eleq2d 2245	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐶 ∈ 𝐵))
39	35, 38	anbi12d 473	. . . . 5 ⊢ (𝑧 = 𝑥 → (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))
40	34, 39	ceqsexv 2774	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
41	33, 40	bitri 184	. . 3 ⊢ (∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
42	24, 41	bitrdi 196	. 2 ⊢ (⟨𝑥, 𝐶⟩ ∈ V → (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))
43	1, 2, 42	pm5.21nii 704	1 ⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1490  [wsb 1760   ∈ wcel 2146  ∃wrex 2454  Vcvv 2735  ⦋csb 3055  {csn 3589  ⟨cop 3592  ∪ ciun 3882   × cxp 4618

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-iun 3884  df-opab 4060  df-xp 4626

This theorem is referenced by:  eliunxp  4759  opeliunxp2  4760  opeliunxp2f  6229