Theorem opeliunxp 5712

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 4950	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)}
2	1	eleq2i 2853	. 2 ⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)})
3		opex 5430	. . 3 ⊢ ⟨𝑥, 𝐶⟩ ∈ V
4		df-rex 3086	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)))
5		nfv 1933	. . . . . 6 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵))
6		nfs1v 2189	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑥 ∈ 𝐴
7		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑧}
8		nfcsb1v 3876	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
9	7, 8	nfxp 5678	. . . . . . . 8 ⊢ Ⅎ𝑥({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
10	9	nfcri 2915	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)
11	6, 10	nfan 1918	. . . . . 6 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
12		sbequ12 2285	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑥 ∈ 𝐴))
13		sneq 4591	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
14		csbeq1a 3866	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
15	13, 14	xpeq12d 5676	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))
16	15	eleq2d 2847	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
17	12, 16	anbi12d 641	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
18	5, 11, 17	cbvexv1 2372	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
19	4, 18	bitri 277	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
20		eleq1 2849	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 𝐶⟩ → (𝑦 ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
21	20	anbi2d 639	. . . . 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 1940	. . . 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 285	. . 3 ⊢ (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵))))
24	3, 23	elab 3638	. 2 ⊢ (⟨𝑥, 𝐶⟩ ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)} ↔ ∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)))
25		opelxp 5681	. . . . . 6 ⊢ (⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑥 ∈ {𝑧} ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵))
26	25	anbi2i 632	. . . . 5 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
27		an12 655	. . . . 5 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
28		velsn 4597	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
29		equcom 2037	. . . . . . 7 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
30	28, 29	bitri 277	. . . . . 6 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
31	30	anbi1i 633	. . . . 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 299	. . . 4 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)))
33	32	exbii 1867	. . 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 2286	. . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
35	14	equcoms 2039	. . . . . . 7 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
36	35	eqcomd 2767	. . . . . 6 ⊢ (𝑧 = 𝑥 → ⦋𝑧 / 𝑥⦌𝐵 = 𝐵)
37	36	eleq2d 2847	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐶 ∈ 𝐵))
38	34, 37	anbi12d 641	. . . 4 ⊢ (𝑧 = 𝑥 → (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))
39	38	equsexvw 2024	. . 3 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
40	33, 39	bitri 277	. 2 ⊢ (∃𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × ⦋𝑧 / 𝑥⦌𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
41	2, 24, 40	3bitri 299	1 ⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798  [wsb 2089   ∈ wcel 2141  {cab 2739  ∃wrex 3085  ⦋csb 3852  {csn 4581  ⟨cop 4587  ∪ ciun 4948   × cxp 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-iun 4950  df-opab 5162  df-xp 5651

This theorem is referenced by:  eliunxp  5807  opeliunxp2  5808  opeliunxp2f  8185  gsum2d2lem  19996  gsum2d2  19997  gsumcom2  19998  dprdval  20028  ptbasfi  23621  cnextfun  24104  cnextfvval  24105  cnextf  24106  dvbsss  25944  iunsnima  32770