Theorem opeliunxp 4735

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 2785	. 2 ⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ⟨𝑥, 𝐶⟩ ∈ V)
2		opexg 4277	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ⟨𝑥, 𝐶⟩ ∈ V)
3		df-rex 2491	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)))
4		nfv 1552	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵))
5		nfs1v 1968	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑥 ∈ 𝐴
6		nfcv 2349	. . . . . . . . . 10 ⊢ Ⅎ𝑥{𝑧}
7		nfcsb1v 3128	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
8	6, 7	nfxp 4707	. . . . . . . . 9 ⊢ Ⅎ𝑥({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
9	8	nfcri 2343	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
10	5, 9	nfan 1589	. . . . . . 7 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
11		sbequ12 1795	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑥 ∈ 𝐴))
12		sneq 3646	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
13		csbeq1a 3104	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
14	12, 13	xpeq12d 4705	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
15	14	eleq2d 2276	. . . . . . . 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 1780	. . . . . 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 2269	. . . . . . 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 1849	. . . . 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 3932	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)}
24	22, 23	elab2g 2922	. . 3 ⊢ (⟨𝑥, 𝐶⟩ ∈ V → (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
25		opelxp 4710	. . . . . . 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 3652	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
29		equcom 1730	. . . . . . . 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 1629	. . . 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 2776	. . . . 5 ⊢ 𝑥 ∈ V
35		sbequ12r 1796	. . . . . 6 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
36	13	equcoms 1732	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
37	36	eqcomd 2212	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ⦋𝑧 / 𝑥⦌𝐵 = 𝐵)
38	37	eleq2d 2276	. . . . . 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 2813	. . . 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 706	1 ⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1516  [wsb 1786   ∈ wcel 2177  ∃wrex 2486  Vcvv 2773  ⦋csb 3095  {csn 3635  ⟨cop 3638  ∪ ciun 3930   × cxp 4678

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 4167  ax-pow 4223  ax-pr 4258

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 3001  df-csb 3096  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-iun 3932  df-opab 4111  df-xp 4686

This theorem is referenced by:  eliunxp  4822  opeliunxp2  4823  opeliunxp2f  6334